Phase Noise of Nanoelectromechanical Sys

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Phase Noise of Nanoelectromechanical Systems
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Yang, Ya-Tang
(2006)
Phase Noise of Nanoelectromechanical Systems.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/YDZP-3X10.
Abstract
Nanoelectromechanical systems (NEMS) are microelectromechanical systems (MEMS) scaled down to nanometer range. As the size of the NEMS resonators is scaled downward, some fundamental and nonfundamental noise processes will impose sensitivity limits to their performance. In this work, we first present theory of phase noise mechanism of NEMS to examine both fundamental and nonfundamental noise processes. Fundamental noise processes considered here include thermomechanical noise, momentum-exchange noise, adsorption-desorption noise, diffusion noise, and temperature-fluctuation noise. For nonfundamental noise processes, we develop a formalism to consider the Nyquist-Johnson noise from transducer-amplifier implementations.
As an initial step to experimental exploration of these noise processes, we describe and analyze several phase-locked loop schemes based on NEMS at very high frequency and ultrahigh frequency bands. In particular, we measure diffusion noise of NEMS arising from xenon atoms adsorbed on the device surface using the frequency modulation phase-locked loop. The observed spectra of fractional frequency noise and Allan deviation agree well with the prediction from diffusion noise theory.
Finally, NEMS resonators also provide unprecedented sensitivity for inertial mass sensing. We demonstrate in situ measurement in real time with mass floor of ~20 zg. Our best mass sensitivity corresponds to ~7 zeptograms, equivalent to ~30 xenon atoms or the mass of an individual 4 kDa molecule. Detailed analysis of the ultimate sensitivity of such devices based on these experimental results indicates that NEMS can ultimately provide inertial mass sensing of individual intact, electrically neutral macromolecules with single-Dalton sensitivity.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
MEMS; nanomechanics; NEMS; phase noise
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Roukes, Michael L.
Thesis Committee:
Roukes, Michael L. (chair)
Scherer, Axel
Goodstein, David L.
Bockrath, Marc William
Defense Date:
9 August 2004
Record Number:
CaltechETD:etd-10162006-124404
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DOI:
10.7907/YDZP-3X10
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06 May 2020 22:42
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PHASE NOISE OF
NANOELECTROMECHANICAL SYSTEMS

Thesis by

Ya-Tang Yang

In Partial Fulfillment of the Requirements for the

Degree of

Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY

Pasadena, California

2007

(Defended August 9, 2004)

Ya-Tang Yang

ill

Dedicated to

my family

iv
ACKNOWLEDGEMENTS
Through the course of my study in Caltech, I have gained much more than I
originally anticipated. During the process of working on graduate research and writing this
thesis, many things have happened that were better and worse than I expect. The period of
time I spent at Caltech certainly brought major transformations to my entire life as a result
of friendship, team work, persistence, intellectual education, sorrow, and merriment.
First, I would like to thank the Lord for His bountiful supply through these years
and I would certainly acknowledge the reigning of His throne over all things through my
toughest time in my personal life and school years. He is active and living God and my
ultimate protector. I sincerely hope this thesis will in some way glorify Him for His purpose
on earth.
I would like to acknowledge the research opportunity provided by my graduate
advisor, Prof. Michael Roukes. Through the training in his research group, I have acquired
incredibly many technical and personal skills that are indispensable in my career
development. I am also privileged to work with so many talented people and benefit from
them. Special thanks go to Prof. Carlo Callegari for his support both on the professional and
the personal level. Carlo has made many breakthroughs in our research sometimes through
our “constructive” confrontation and brainstorming. I very much enjoyed his company for
the time we spent chatting about many small but important things about life in the cafe and
the lab. I would also like to thank Philip Xiao-li Feng for his friendship, hands-on help in the
lab, and efforts to get our results published. I have collaborated on various projects with
Prof. Kamil Ekinci, Larry Schiviaone, Dr. Warren Fon, Dr. Darrel Harrington, Dr. Henry
Xue-Ming Huang, Dr. Hong Tang, Dr. Ali Husain, Prof. Chris Zorman, Prof. Jim Hone, Dr.
Henk Postma, Prof. Keith Schwab, Meher Prakash, Inna Kozinsky, and many others, whose

help has been invaluable for making my graduate research productive. I also thank our
honorary group member, Prof. Philps Wigen from Ohio State University, for his personal
support and encouragement.
I am grateful to my mother Li-chu Chu for her unconditioned love and support and
dedicate my doctorate degree to her. During my stay at Caltech, major tragedy has happened
to my family. Ever since, the prayer, fellowship, support, and practical advice from the
brothers and sisters of the church in Monterey Park and Santa Clara have eased my pain and
anxiety in this prolonged academic process of obtaining the diploma. Last but not least, I
would like thank my fiancée, Jessie Yung-chieh Yu, for her thoughtfulness and prayer for me
and my family. She has brought me joy and peace beyond words and knowledge and
dissolved my confusion about life in many cases. As we begin the new phase in our life, I am
indebted to her for her love through this period and for many years to come.

vi
ABSTRACT
Nanoelectromechanical systems (NEMS) are microelectromechanical devices
(MEMS) scaled down to nanometer range. NEMS resonators can be fabricated to achieve
high natural resonance frequencies, exceeding 1 GHz with quality factors in excess of 104.
These resonators are candidates for ultrasensitive mass sensors and frequency determining
elements of precision on-chip clocks. As the size of the NEMS resonators is scaled
downward, some fundamental and nonfundamental noise processes will impose sensitivity
limits to their performance. In this work, we examine both fundamental and
nonfundamental noise processes to obtain the corresponding expressions for phase noise
density, Allan deviation, and mass sensitivity. Fundamental noise processes considered here
include thermomechanical noise, momentum-exchange noise, adsorption-desorption noise,
diffusion noise, and temperature-fluctuation noise. For nonfundamental noise processes, we
develop a formalism to consider the Nyquist-Johnson noise from transducer-amplifier
implementations.
As an initial step to experimental exploration of these noise processes, we
demonstrate the phase noise measurement of NEMS using the phase-locked loop scheme.
We analyze control servo behavior of the phase-locked loop and describe several
implementation schemes at very high frequency and ultra high frequency bands. By
incorporating the ~190 MHz NEMS resonator into the frequency modulation phase-locked
loop, we investigate the diffusion noise arising from xenon atoms adsorbed on the device
surface. Our experimental results can be explained with the diffusion noise theory. The
measured spectra of fractional frequency noise confirm the predicted functional form from
the diffusion noise theory and are fitted to extract the diffusion coefficients of adsorbed
xenon atoms. Moreover, the observed Allan deviation is consistent with the theoretical

vii
estimates from diffusion noise theory, using the total number of adsorbed atoms and
extracted diffusion times.
Finally, very high frequency NEMS devices provide unprecedented potential for
mass sensing into the zeptogram level due to their minuscule mass and high quality factor.
We demonstrate in situ measurements in real time with mass noise floor ~20 zeptogram.
Our best mass sensitivity corresponds to ~7 zeptograms, equivalent to ~30 xenon atoms or
the mass of an individual 4 kDa molecule. Detailed analysis of the ultimate sensitivity of
such devices based on these experimental results indicates that NEMS can ultimately provide
inertial mass sensing of individual intact, electrically neutral macromolecules with singleDalton (1 amu) sensitivity. This is an exciting prospect—when realized it will blur the
traditional distinction between inertial mass sensing and mass spectrometry. We anticipate
that it will also open intriguing possibilities in atomic physics and life science.

viii
CONTENTS
Acknowledgements............................................................................................................iv
Abstract...............................................................................................................................vi
Table of Contents............................................................................................................ viii
List of Figures .....................................................................................................................x
List of Tables.................................................................................................................... xii
Chapter 1 Overview ...................................................................................................................................1
1.1 Nanoelectromechancial Systems ..............................................................................................1
1.2 Brownian Motion, Nyqusist-Johnson Noise, and Fluctuation-Dissipation Theorem
........................................................................................................................................................2
1.3 Noise in Microelectromechancial Systems and Nanoelectromechanical Systems
........................................................................................................................................................3
1.4 Phase Noise in Microelectromechancial Systems and Nanoelectromechanical Systems
.........................................................................................................................................................5
1.5 Mass Sensing Based on Microelectromechancial Systems and Nanoelectromechanical
Systems ..........................................................................................................................................7
1.6 Organization ................................................................................................................................8
Chapter 2 Introduction to Phase Noise ............................................................................................... 13
2.1 Introduction .............................................................................................................................. 14
2.2 General Remark......................................................................................................................... 14
2.3 Phase Noise ................................................................................................................................ 15
2.4 Frequency Noise........................................................................................................................ 17
2.5 Allan Variance and Allan Deviation ...................................................................................... 18
2.6 Thermal Noise of an Ideal Linear LC Oscillator................................................................. 22
2.7 Minimum Measurable Frequency Shift ................................................................................. 25
2.8 Conclusion.................................................................................................................................. 26
Chapter 3 Theory of Phase Noise Mechanisms of NEMS .............................................................. 29
3.1 Introduction .............................................................................................................................. 30
3.2 Thermomechanical Noise........................................................................................................ 33
3.3 Momentum Exchange Noise .................................................................................................. 37
3.4 Adsorption-Desorption Noise................................................................................................ 38
3.5 Diffusion Noise ......................................................................................................................... 50
3.6 Temperature Fluctuation Noise ............................................................................................ 57

ix
3.7 Nonfundamental Noise............................................................................................................ 59
3.8 Conclusion.................................................................................................................................. 61
Chapter 4 Experimental Measurement of Phase Noise in NEMS ................................................ 67
4.1 Introduction .............................................................................................................................. 68
4.2 Analysis of Phase-Locked Loop Based on NEMS ............................................................. 69
4.3 Homodyne Phase-Locked Loop Based upon a Two-Port NEMS Device.................... 77
4.4 Frequency Modulation Phase-Locked Loop........................................................................ 84
4.5 Comparison with Local Oscillator Requirement of Chip Scale Atomic Clock ............. 96
4.6 Experimental Measurement of Diffusion Noise............................................................... 100
4.7 Conclusion................................................................................................................................ 114
Chapter 5 Zeptogram Scale Nanomechanical Mass Sensing ....................................................... 119
5.1 Introduction ........................................................................................................................... 120
5.2 Experimental Setup .............................................................................................................. 120
5.3 Mass Sensing at Zeptogram Scale ...................................................................................... 124
5.4 Conclusion............................................................................................................................... 128
Chapter 6 Monocrystalline Silicon Carbide Nanoelectromechanical Systems ......................... 131
6.1 Introduction ........................................................................................................................... 132
6.2 Device Fabrication and Measurement Results ................................................................. 134
6.3 Conclusion............................................................................................................................... 141
Chapter 7 Balanced Electronic Detection of Displacement of Nanoelectromechanical
Systems ............................................................................................................................... 145
7.1 Introduction ........................................................................................................................... 146
7.2 Circuit Scheme and Measurement Results ....................................................................... 146
7.3 Conclusion............................................................................................................................... 155

FIGURES
2.1 Definition of phase noise ................................................................................................................. 16
2.2 Plot of the function F (x ) ................................................................................................................ 21
2.3 Leeson’s model of phase noise for an ideal linear LC oscillator............................................... 24
2.4 Summary of the relation between different quantities................................................................ 27
3.1 Vibrational mode shape of the beam with doubly clamped boundary condition imposed
and its Gaussian approximation...................................................................................................... 54
3.2 Plot of the function ξ (x) . ............................................................................................................... 55
3.3 Plot of Χ (x ) and its asymptotic form.......................................................................................... 56
4.1 Self-oscillation scheme for the phase noise measurement of NEMS...................................... 70
4.2 Configuration of a phase-locked loop based on NEMS ........................................................... 71
4.3 Pictures of two-port NEMS devices. ............................................................................................. 79
4.4 Implementation of the homodyne phase-locked loop based on a two-port NEMS device
...............................................................................................................................................................80
4.5 Mechanical resonant response after nulling.................................................................................. 81
4.6 Phase noise density of the 125 MHz homodyne phase-locked loop based on a two-port
NEMS device. .................................................................................................................................... 82
4.7 Allan deviation of the 125 MHz homodyne phase-locked loop based on a two-port NEMS
device. ................................................................................................................................................... 83
4.8 Conceptual diagram of frequency modulation phase-locked loop (FM PLL) scheme......... 85
4.9 Implementation of frequency modulation phase-locked loop (FM PLL) scheme.................88
4.10 Phase noise density of the 190 MHz frequency modulation phase-locked loop (FM PLL)
…................................................................................................................................................................. 92
4.11 Allan deviation of the 133 MHz frequency modulation phase-locked loop (FM PLL)….93
4.12 Phase noise density of the 419 MHz frequency modulation phase-locked loop (FM PLL)
.......................................................................................................................................................….94
4.13 Allan deviation of the 419 MHz frequency modulation phase-locked loop (FM PLL)
..............................................................................................................................................................95

xi
4.14 Phase noise spectrum of NEMS-based phase-locked loops versus the local oscillator (LO)
requirement of chip scale atomic clock (CSAC)........................................................................98
4.15 Allan deviations of NEMS-based phase-locked loops versus the local oscillator (LO)
requirement of chip scale atomic clock (CSAC)........................................................................99
4.16 Experimental configuration for diffusion noise measurement.............................................102
4.17 Adsorption spectrum of xenon atoms on NEMS surface.....................................................104
4.18 Representative fractional frequency noise spectra...................................................................106
4.19 Spectral density of fractional frequency noise contributed from gas...................................107
4.20 Spectral density of fractional frequency noise with fitting......................................................110
4.21 Allan deviation data with gas and without gas..........................................................................112
4.22 Comparison with prediction from diffusion noise theory and Yong and Vig’s model....113
5.1 Experimental configuration. .......................................................................................................... 123
5.2 Real time zeptogram-scale mass-sensing experiment................................................................ 126
5.3 Mass responsivities of nanomechanical devices......................................................................... 127
6.1 SEM picture of doubly clamped SiC beams. .............................................................................. 136
6.2 Representative data of mechanical resonance.............................................................................138
6.3 Frequency versus effective geometric factor for three families of doubly clamped beams
made from single-crystal SiC, Si, and GaAs ............................................................................... 141
7.1 Schematic diagrams for the magnetomotive reflection measurement and bridge
measurement .................................................................................................................................... 147
7.2 Data from a doubly clamped n+ Si beam .................................................................................... 151
7.3 Narrowband and broadband transfer function from metalized SiC beam in bridge
configuration ................................................................................................................................... 153

xii
TABLES
3.1 Allan deviation and mass sensitivity limited by thermomechanical noise for representative
realizable NEMS device configurations ......................................................................................... 36
3.2 Summary of Yong and Vig’s and ideal gas models...................................................................... 48
3.3 Maximum Allan deviation and mass fluctuation of representative NEMS devices.............. 49
3.4 Summary of expressions for spectral density and Allan deviation for fundamental noise
processes considered in this work. ................................................................................................. 64
4.1 Summary of parameters of all phase-locked loops based on NEMS presented in this work
............................................................................................................................................................... 76
4.2 Summary of experimental parameters used in the frequency modulation phase-locked
loops (FM PLL) at very high frequency (VHF) and ultra high frequency (UHF) bands.....91
4.3 Summary of diffusion times and coefficients versus temperature..........................................111

Chapter 1
Overview

1.1 Nanoelectromechanical Systems
Nanoelectromechanical systems (NEMS) are microelectromechanical devices
(MEMS) scaled down to nanometer range.1 NEMS have a lot of intriguing attributes.2
They offer access to fundamental frequencies in the microwave range; 3 quality factor (Q)
in the tens of thousands; 4 active mass in the femtogram range; force sensitivities at the
attonewton level;5,6 mass sensitivity at the level of individual molecules7 — this list goes
on. These traits translate into new prospects for a variety of important technological
applications. Among them, nanomechanical resonators are rapidly being pushed to
smaller sizes and higher frequencies due to their applications as Q filters and on-chip
clocks.4 The fully integrated NEMS oscillators will boast smaller size and lower power
consumption and thus can potentially replace their macroscopic counterparts such as the
quartz crystal oscillators and surface wave acoustic resonators.
The resonance frequency in general scales as 1/L, where L is the scale of the
resonator. As size scales are reduced and frequency is increased, the corresponding
statistical fluctuations will be more pronounced and inevitably limit performance. The
central question of this thesis is: as the size of the resonator becomes smaller, how stable

can the resonant frequency be? The answers to this seemingly simple question form the
subject of phase noise of NEMS. We will review the pioneering work before going to this
subject in detail.

1.2 Brownian Motion, Nyquist-Johnson Noise, and Fluctuation-Dissipation
Theorem
A microscopic particle immersed in a liquid exhibits a random type of motion.
This phenomenon is called Brownian motion and reveals clearly the statistical
fluctuations that occur in a system in thermal equilibrium.8 The Einstein relation, perhaps
the most important result of the study of Brownian motion, states that the diffusion
constant is proportional to the frictional coefficient determined by the hydrodynamic
interaction of the particle with the viscous fluid.12 The Brownian motion serves as a
prototype problem whose analysis provides considerable insight into the mechanisms
responsible for the existence of fluctuations and dissipation of energy. This problem is
also of great practical interest because such fluctuations constitute a background of
“noise” which imposes sensitivity limits on delicate physical measurements. For
example, Nyquist-Johnson noise, which originates from thermal agitation of electrical
charge in a conductor,9,10 is present at any circuitry with nonzero dissipation, and in many
cases determines the noise floor of an amplifier.11 Nyquist’s theorem states that the
spectral density of the thermal fluctuating voltage of any electrical impedance is always
proportional to the square root of its resistive part.13 The same arguments used to study
Brownian motion and Nyquist-Johnson noise can be extended on a more abstract level to
a general result of wide applicability, the fluctuation-dissipation theorem.12-14 The

fluctuation-dissipation theorem explicitly indicates how the cross-correlation functions of
the fluctuating quantities are associated with the friction coefficients of the equations of
motion, or equivalently, how the spectra of statistical fluctuations are related to the
dissipations of the system near thermal equilibrium.

1.3 Noise in Microelectromechancial Systems and Nanoelectromechanical
Systems
We now review the study of the noise of MEMS and NEMS, starting from the
work in a liquid. Paul and Cross have considered the Brownian motion of NEMS
cantilevers and concluded that the corresponding force sensitivities are in the range of
piconewton.15 Considering the hydrogen bond strength is ~10 pN, such sensitivities
imply the possibility of using NEMS to sense biological forces at single molecule level.
On the other hand, optical tweezers have recently led to quite spectacular measurements
of small weak force, with the force sensitivities again limited by Brownian motion.16 In
this technique, an optical beam, focused to the diffraction limit, is employed.
Functionalized dielectric beads, typically having diameters of ~1 μm, are attached to the
biomolecules under study to provide a handle. In this way, direct measurements of
piconetwon scale biological forces have been obtained.18 In a more recent study, internal
dynamics of DNA, yielding forces in the femtonewton range, have been observed via the
two-point correlation technique.17
We now discuss the work on characterization the thermomechancial noise of
MEMS and NEMS in vacuum. Albrecht et al. demonstrate frequency modulation
detection using high Q cantilevers for enhanced force microscopy sensitivity, limited by

thermomechancial noise in vacuum.19 Similarly, using a high Q single crystal silicon
cantilever as thin as 60 nm, T. D. Stowe et al. have achieved attonewton force sensitivity
at 4.8 K in vacuum.5 Cooling down similar devices further to millikelvin temperatures,
force sensitivity at subattonewton scale has also been demonstrated.6 Such exquisite force
sensitivities have ultimately led to the detection of single electron spin using magnetic
resonance force microscopy (MRFM).20
The observation of thermomechancial noise of high frequency NEMS has been
hindered, largely due to the diminishing transducer responsivity as the dimensions are
reduced into the submicron range. This can only be circumvented by delicate
incorporation of the actuator, transducer, and readout amplifier, all meticulously chosen
and orchestrated to minimize the noise from these extrinsic elements. For example, the
piezoresistors on NEMS silicon cantilevers, which acts as transducers upon current
biasing, convert the mechanical displacement into a voltage signal, which is subsequently
read out by a low noise amplifier. Using such a scheme, Arlett et al. have observed the
theromomechanical noise down to cryogenic temperatures for NEMS devices with
resonance frequencies of ~2 MHz.22
Another example is the nanomechanical parametric amplifier at 17 MHz by
Harrington,23 which is similar to the one demonstrated by Rugar and Grutter using a
microscale cantilever.21 Operating in degenerate mode, a parametric modulation of the
beam’s effective stiffness at twice the signal frequency is produced by the application of
an alternating longitudinal force to both ends of a doubly clamped beam. At highest
mechanical gains, noise matching performance is achieved, resulting in the observation
of thermomechanical noise squeezing at cryogenic temperatures.

Finally, we mention the recent attempt to approach the quantum limit of a
nanomechancial resonator by coupling a single electron transistor (SET) with a high Q,
19.7 MHz nanomechanical resonator by LeHaye et al.24 At temperatures as low as 56
millikelvin, they observe thermomechanical noise corresponding to a quantum
occupation number of 58, and demonstrate the near-ideal performance of the SET as a
linear amplifier. This work clearly paves the feasible way to the quantum mechanical
limits of NEMS, blurring the division between quantum optics and solid state physics.2

1.4

Phase

Noise

Microelectromechancial

Systems

and

Nanoelectromechanical Systems
We now review work on the phase noise of NEMS and MEMS. The phase noise
of MEMS resonators was first analyzed by Vig and Kim.25 They examine how frequency
stabilities of MEMS and NEMS resonators scale with dimensions. When the dimensions
of a resonator becomes small, instabilities that are negligible in macroscale devices
become prominent. At submicron dimensions, the temperature fluctuation noise,
adsorption-desorption noise, and thermomechanical noise are likely to limit the
applications of ultra small resonators. Later, Cleland and Roukes develop a selfcontaining formalism to treat a similar list of noise sources and estimate their impact on a
doubly clamped beam of single crystal silicon with a resonance frequency of 1 GHz.26
Their calculation, however, does not agree with Vig and Kim’s work in terms of the
magnitude of the impact of the noise, as well as the method of analysis of some of the
noise sources, in particular, that of the effect of temperature fluctuations. In analyzing the
temperature fluctuation noise, they consider a more realistic thermal circuit by dividing

the device into sections, and show that the resulting Allan variance is of the same
magnitude as that due to thermomechanical noise for the model resonator with Q of 104.
This apparently contradicts the excessive temperature fluctuations predicted by Vig and
Kim.25 Moreover, they conclude that the noise performance, limited by the fundamental
noise processes, can be comparable with their macroscale counterparts, the oven
stabilized quartz crystal oscillators. By consolidating these studies, we first introduce the
subject of phase noise in chapter 2, and then present the theory of the phase noise
mechanisms affecting NEMS in chapter 3.
Except for the aforementioned theoretical works, very little experimental data are
available for evaluating whether the calculated noise performance can be achieved. More
systematic approaches, measuring the performance of high Q resonators operating in
phase-locked loops, with controlled variations in temperature, environment, and
materials, need to be followed. As an initial step into these efforts, we describe the
implementations of phase-locked loops based on NEMS devices in chapter 4. We also
report the observation of adsorption-desorption noise arising from xenon atoms adsorbed
on the device surface. Our measurement results are in excellent agreement with the
proposed idea gas model. More generally, our approach represents a canonical example
on how to study the frequency stabilities arising from a particular noise process of
interest.

1.5 Mass Sensing Based on Microelectromechanical Systems and
Nanoelectromechancial Systems
We now review a separate, but closely related front: the inertial mass sensing
based on MEMS and NEMS. Today mechanically based sensors are ubiquitous, having a
long history of important applications in many diverse fields of science and technology.
Among the most responsive sensors are those based on the acoustic vibratory modes of
crystals,27,28 thin films,29 and more recently, MEMS30,31 and NEMS.7,32, Three attributes
of these devices establish their mass sensitivity: effective vibratory mass, quality factor,
and resonant frequency. The miniscule mass, high Q, and high resonant frequency of
NEMS provide them with unprecedented potential for mass sensing. Femtogram mass
sensing using NEMS cantilevers has been demonstrated by Lavrik and Datskos by
photothermally exciting silicon cantilevers in the range of 1 to 10 MHz and measuring a
mass change of 5.5 fg upon chemisorption of 11-mercaptoundecanoic acid.32 Ekinci and
Roukes achieve attogram mass sensing by exposing NEMS devices with Au atomic flux
and tracking the resulting frequency shift in a phase-locked loop.33 Motivated by these
experiments, we start to examine theoretically the ultimate limits of inertial mass sensing
based upon NEMS devices as a result of fundamental noise processes.7 We present the
resulting theoretical analysis in chapter 3. The conclusion is quite compelling: it indicates
that NEMS devices can directly “weigh” individual intact, electrically neutral, molecules
with single Dalton sensitivities.
As an initial step toward this goal, we present our mass sensing experiments at
zeptogram scale in chapter 4. This is demonstrated by depositing xenon atoms and
nitrogen molecules on the NEMS device, and tracking the resulting frequency shift in

high precision phase-locked loop. But more importantly, the agreement of our
experimental results with the theory justifies our formalism and validates its use to
delineate, for the first time, the feasible pathway into single Dalton sensitivity.

1.6 Organization
To help the reader understand this work in a more coherent and clear way, this
thesis is organized in the following way:
Chapter 2 introduces the subject of phase noise and serves as the mathematical
foundation of this work. We first describe how phase fluctuations of an oscillator convert
to the noise sideband of the carrier. We then define the phase noise, the frequency noise,
and Allan deviation, emphasizing their relationship with each other. As an example,
Leeson’s model is described and used to analyze the thermal noise of an ideal linear LC
oscillator.
Chapter 3 discusses the phase noise mechanism of the NEMS resonators. We first
examine fundamental noise processes, including thermomechancial noise, momentum
exchange noise, adsorption-desorption noise, diffusion noise, and temperature fluctuation
noise. We also discuss nonfundamental noise processes arising from the Nyquist-Johnson
noise of the transducer amplifier implementations. For each noise process presented here,
we give expressions for the phase noise spectra and Allan deviation and then translate
them into the corresponding minimum measurable frequency shift and mass sensitivity in
light of their importance in sensing applications.
Chapter 4 presents the experimental measurement of the phase noise of NEMS.
First, we first analyze the control servo behavior of the phase-locked loops and give the

detailed implementations together with their noise performance. The achieved noise
performance is compared to the local oscillator (LO) requirements of chip scale atomic
clocks (CSAC) to evaluate the viability of NEMS based oscillators for this application.
Finally, we investigate the diffusion noise arising from adsorbed xenon atoms by putting
a very high frequency NEMS into the phase-locked loop and measuring the frequency
noise spectra and Allan deviation.
Chapter 5 shows very high frequency NEMS that provide a profound sensitivity
increase for inertial mass sensing into zeptogram scale. We demonstrate real time, in situ
mass detection of sequential pulses of ~100 zg nitrogen molecules by tracking resulting
frequency shift. Measurement and analysis from our experiments demonstrate mass
sensitivities at the level of ~7 zg, the mass of an individual 4 kDa molecule, or ~30 xenon
atoms.
Chapter 6 describes a surface nanomachining process that involves electron beam
lithography, followed by dry anisotropic and selective electron cyclotron resonance
plasma etching steps. Measurements on a representative family of the resulting devices
demonstrate that, for a given geometry, nanometer-scale SiC resonators are capable of
yielding substantially higher frequencies than GaAs and Si resonators.
Chapter 7 describes a broadband radio frequency balanced bridge technique for
electronic detection of displacement in NEMS. The effectiveness of the technique is
demonstrated by detecting the minute electromechanical impedances of NEMS
embedded in large electrical impedances at very high frequencies.

References

M. L. Roukes Plenty of room indeed. Sci. Am. 285, 48 (2001).

M. L. Roukes Nanoelectromechanical Systems. Technical Digest, Solid State
Sensor and Actuator Workshop, Hilton Head Island, South Carolina (June4–8,
2000) 367–376 (2000).

X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes Nanodevices
motion at microwave frequencies. Nature 421, 496 (2003).

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mechanical resonators from bulk Si crystals Appl. Phys. Lett. 69, 2653 (1996).

T. D. Stowe, K. Yasumura, T.W. Kenny, D. Botkin, K. Wago, and D. Rugar
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motion of two colloidal spheres. J. Chem. Phys. 106, 2837 (1997).

17.

J. C. Meiners and S. R. Quake Femtonewton force spectroscopy of single
extended DNA molecules. Phys. Rev. Lett. 84, 5014 (2000).

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force clamp. Biophysical Journ. 74, A49 (1998).

19.

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detection using high Q cantilever for enhanced force microscopy sensitivity J.
Appl. Phys. 69, 668 (1991).

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D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui Single spin detection by
magnetic resonance force microscope. Nature 430, 329 (2004).

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thermomechanical noise squeezing. Phys Rev. Lett. 67, 699 (1991).

22.

J. L. Arlett, J. R. Maloney, B. Gudlewski, M. Muluneh, and M. L. Roukes Selfsensing micro- and nanocantilevers with attonewton-scale force resolution. Nano
Lett. 6, 1001, (2006).

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D. A. Harrington Physics and applications of nanoelectromechanical systems
(NEMS). PhD thesis, California Institute of Technology (2003).

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quantum limit of a nanomechanical resonator. Science 304, 74 (2004).

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11

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Chapter 2
Introduction to Phase Noise

A brief introduction into the subject of phase noise is given here. We first describe
the conversion of the phase fluctuations into the noise sideband of the carrier. We
then define phase noise, frequency noise, and Allan deviation with emphasis on their
relationship with each other. Leeson’s model is described and used to analyze the
thermal noise of an ideal, linear LC oscillator. Finally, we give the general
expression of the minimum measurable frequency shift in a noisy system.

2.1 Introduction
In general, circuit and device noise can perturb both the amplitude and phase of
an oscillator’s output.1,2 Of necessity, all practical oscillators inherently possess an
amplitude limiting mechanism of some kind. Because the amplitude fluctuations are
attenuated, phase noise generally dominates. We will primarily focus on phase noise in
our theoretical exposition and divide the theoretical investigation into two parts. The first
part is the general conceptual foundation on how the frequency stability of an oscillator
should be characterized, more commonly known as the subject of phase noise. The
second part is the exposition on the physical phase noise mechanisms affecting NEMS
devices. In this chapter, we will deal with the first part and defer the second part to
chapter 3. We will also describe Leeson’s model to analyze the thermal noise of an ideal,
linear LC oscillator. Finally, we will give expressions translating the frequency noise into
the minimum measurable frequency shift in a noisy system.

2.2 General Remark
The output of an oscillator of angular frequency ωC is generally given by
X (t ) = X 0 (1 + A(t )) f [ωC t + φ (t )] .

(2.1)

Here φ (t ) and A(t ) are functions of time and f is a periodic function. Here X can
be the output voltage from an electrical oscillator or the displacement of a mechanical
oscillator. The output spectrum contains higher harmonics of ωC if the waveform is not
sinusoidal. For our purpose, we assume no higher harmonics from any nonlinearity of the
devices or the circuits, and thus the output X (t ) is purely sinusoidal. For a sinusoidal
oscillation, the output is given by
14

X (t ) = X 0 (1 + A(t )) sin[ω C t + φ (t )] .

(2.2)

2.3 Phase Noise
The physical fluctuations in the oscillator can perturb the phase of the oscillation
and produce phase fluctuations. We now describe how then phase fluctuations are
converted into noise sidebands around the carrier. Considering a small phase
variation φ (t ) = φ 0 sin ωt , equation (2.2) can be expanded as
X (t ) = X 0 (1 + A(t )) sin(ω C t + φ 0 sin ωt + θ )
= X 0 sin(ω C t + θ ) + X 0

φ0

sin[(ω C + ω )t ] − X 0

φ0

sin[(ω C − ω )t ].

(2.3)

The phase variation generates two sidebands spaced ±ω from the carrier with
amplitude X0φ0 / 2. The upper sideband is phase-coherent with the lower sideband with
the opposite sign. The generated sideband is characterized in the following definition: it
is conventionally given by the ratio of noise power to carrier power for 1 Hz bandwidth
with offset frequency from the carrier. In notation, the definition is given by
⎛P
(ω + ω ,1Hz ) ⎞
⎟⎟ .
Ltotal (ω ) = 10 log⎜⎜ sideband C
PC

(2.4)

PC is the carrier power and Psidebank (ω C + ω ,1Hz ) is the single sideband power at a
frequency offset ω from the carrier frequency ωC with the measurement bandwidth of 1
Hz as shown in figure 2.1. Ltotal (ω) is thus in units of decibel referred to the carrier
power per hertz (dBc/Hz).

dBc

ωc 1Hz

Figure 2.1. Definition of phase noise. The phase noise is conventionally expressed as
the ratio of sideband noise power for 1 Hz bandwidth to the carrier power in units of
dBc/Hz.

2.4 Frequency Noise
Phase is the integration of frequency over time, i.e.,

φ (t ) = ∫ ω (τ )dτ .

(2.5)

−∞

Conversely, frequency is the derivative of phase with respect to time, i.e.,

ω (t ) =

dφ
dt

(2.6)

The spectral density of the phase noise is thus related to the spectral density of the
frequency noise by
Sφ (ω ) =

ω2

S ω (ω ) .

(2.7)

In addition to angular frequency, we introduce another commonly used quantity,
fractional frequency, defined as ratio of frequency to carrier frequency.
y=

δω
ωC

(2.8)

The spectral density of fractional frequency is related to the spectral density of frequency
by
S y (ω ) =

ω C2

S ω (ω ) .

(2.9)

The resonance frequency depends on many physical parameters of the resonator.
The fluctuations of these parameters can translate into fractional frequency noise. The
fractional noise is related to the fluctuation of the corresponding parameter by

⎛ ∂y ⎞
S y (ω ) = ⎜⎜ ⎟⎟ S χ .
⎝ ∂χ ⎠

(2.10)

χ is the physical parameter which the resonant frequency is dependent on. For example,
if χ is the temperature T of the device, ∂y / ∂T is simply the temperature coefficient of

the resonant frequency.

2.5 Allan Variance and Allan Deviation

Allan variance is a quantity commonly used by the frequency standard
community to compare the frequency stabilities of different oscillators. The phase and
frequency noise are defined in the frequency domain; the Allan deviation is defined in the
time domain. Allan deviation, σ A (τ A ) , is simply the square root of Allan variance,

σ A2 (τ A ) . The defining expression of the Allan deviation is given by1,3
σ A2 (τ A ) =

1 NS
( f m − f m −1 ) 2 .
2 f C N − 1 m=2

(2.11)

f m is the average frequency measured over the mth interval with zero dead time

and N S is the sample number. From this definition, the Allan deviation is related to the
phase noise density by

⎛ 2 ⎞ ∞
⎟⎟ ∫ S φ (ω ) sin 4 (ωτ A / 2)dω .
σ (τ A ) = 2⎜⎜
⎝ ωτ A ⎠ 0

(2.12)

In the experimental data, Allan deviation is usually presented with the error bar
given by one standard deviation confidence interval (or 68% confidence interval), i.e.,

σ A / N S − 1 . For example, for sample number NS=101, the one standard deviation
confidence interval is 10% of the Allan deviation.

The noise spectra with different power laws are commonly used so we give the
formulas of the corresponding Allan deviations. For phase noise having 1 / f 4
component, i.e., S φ (ω ) = C 4 (ω C / ω ) 4 , the Allan deviation is given by

σ A (τ A ) =

C 4ω C2τ A .

(2.13)

For phase noise having 1 / f 3 component, i.e., S φ (ω ) = C 3 (ω C / ω ) 3 , the Allan deviation is
given by

σ A (τ A ) = 2 log e 2C 3ω C .

(2.14)

For phase noise having 1 / f 2 component, i.e., S φ (ω ) = C 2 (ω C / ω ) 2 , the Allan deviation
is given by

σ A (τ A ) =

πC 2
τA

(2.15)

For the fractional frequency noise having the Lorentizian function form, i.e.,
S y (ω ) = A /(1 + (ωτ r ) 2 ) ,

the

spectral

density

phase

noise

given

S φ (ω ) = A(ω C / ω ) 2 /(1 + (ωτ r ) 2 ) . Upon integration, the Allan deviation is given by

F( r ) .
2π τ A

σ A (τ A ) =

(2.16)

F (x) is an analytic function defined by

F ( x) =

1 sin 4 (ξx / 2)dξ
− 2 [(1 − e − x ) − (1 − e − 2 x )] .
2 ∫
2x x
x 0 ξ (1 + ξ )

(2.17)

As shown in figure 2.2, F (x) reaches a maximum at x=1.89 with the value 0.095. The
asymptotic expressions of F (x) are F ( x) =

for x>>1 and F ( x) = x for x <<1.
2x
19

These behaviors can also be clearly seen in figure 2.2. In the limit τ r << τ A , equation
(2.16) becomes

σ A (τ A ) =

Aτ r
4πτ A

(2.18)

In the other limit τ A << τ r , equation (2.16) becomes

σ A (τ A ) =

Aτ A
12πτ r

(2.19)

-1

-2

-3

F(x)

F( x ) ∝ x

-3

-2

-1

F( x ) ∝ 1/ x

Figure 2.2. Plot of the function F(x). F(x) shows the dependence of Allan deviation,

having frequency noise density of Lorentzian form, on the ratio of the correlation time

τ r to the averaging time τ A . F (x) reaches a maximum at x=1.85 with the value 0.095.
Its asymptotic behaviors for x <<1 and for x>>1 are also shown.

2.6 Thermal Noise of an Ideal Linear LC Oscillator

The phase noise of an ideal linear LC oscillator due to the Nyquist-Johnson noise
is analyzed by Leeson.4 Figure 2.3 shows that the Nyquist-Johnson noise source
associated with the resistor injects noise current into a LC tank circuit. The impedance of
the LC tank with a quality factor Q and the resonant frequency ω 0 at offset frequency ω
( ω << ω 0 ) is given by
Z (ω 0 + ω ) =

1 + j 2Q
ω0

(2.20)

To sustain oscillation, the active device must compensate the energy dissipation
by positive feedback. Therefore, the active device behaves as a negative conductance
− G . For steady state oscillation, the impedance of the oscillator model is given by

Z (ω ) =

vout (ω 0 + ω )
1 ω0
=−j
iin (ω 0 + ω )
G 2Qω

(2.21)

The total equivalent parallel resistance of the tank has an equivalent mean square
noise current density of iin2 / Δf = 4k B TG . Using this effective current power, the phase

noise can be calculated as

Sφ (ω ) =

noise
signal

Z (ω ) iin2 / Δf
k BT ⎛ ω 0 ⎞
⎜ ⎟ .
1 2
2 PC Q 2 ⎝ ω ⎠
Vo

(2.22)

PC is the carrier power usually limited by saturation or nonlinearity of the active device.
The Leeson model demonstrates explicitly the conversion of the current noise into
sideband and explains the 1 / ω 2 dependence of the phase noise density. Upon integration
of the spectral density, we obtain the expression for the Allan deviation.
22

σ A (τ A ) =

k BT 1
PC Q 2τ A

(2.23)

Active Device

i (ω)

-G

Figure 2.3. Leeson’s model of phase noise for an ideal linear LC oscillator.

Equivalent one-port circuit for phase noise calculation for an ideal linear LC oscillator is
used in the model. The Nyquist-Johnson noise source associated with the resistor injects
noise current in LC tank, producing the noise sideband around the carrier. Note that the
active device, compensating the energy dissipation from the resistor, is modeled as a
negative conductance.

2.7 Minimum Measurable Frequency Shift

Experimentally we measure the change in physical properties of the resonator by
detecting the corresponding frequency shift and thus an important question needs to be
addressed:what is the minimum measurable frequency shift, δω 0 , that can be resolved in
a (realistic) noisy system? In principle, a shift comparable to the mean square noise (the
spread) in an ensemble average of a series of frequency measurements should be
resolvable, i.e., δω0 ≈

∑ (ω − ω )
i =1

for signal-to-noise ratio equal to unity. An

estimate for δω 0 can be obtained by integrating the weighted effective spectral density of
the frequency fluctuations, S ω (ω ) , by the normalized transfer function of the
measurement loop, H (ω ) :

δω 0 ≈ [ ∫ S ω (ω ) H (ω )dω ]1 / 2 .

(2.24)

Here, S ω (ω ) is in units of (rad / s 2 ) /(rad / s) . We can further simplify equation
(2.24) by replacing H (ω ) with the square transfer function H ' (ω ) , which has the same
integrated spectral weight, but is non-zero only within the passband delineated by 2πΔf .
Here, Δf ≈ 2π / τ and is dependent upon the measurement averaging time, τ. Given this
assumption, equation (2.24) takes the simpler, more familiar form.
2πΔf

δω0 ≈ [ ∫ S ω (ω )dω ]1 / 2 .

(2.25)

This, of course, is an approximation to a real system — albeit a good one. If necessary,
one can resort to the more accurate expression, equation (2.24).

2.8 Conclusion

We describe the conversion of phase fluctuations into the noise sideband of the
carrier and present the definitions of phase noise, frequency noise, and Allan deviation,
all commonly used to characterize the frequency stability of an oscillator. Figure 2.4
summarizes the relation between these quantities. We illustrate these definitions by
analyzing the phase noise of an ideal, linear LC oscillator in the context of Leeson’s
model. In particular, Leeson’s model explicitly demonstrates how the Nyquist-Johnson
current noise produces noise sideband of carrier and explains the 1/ ω 2 dependence of the
phase noise density on the offset frequency. Finally, we give the expressions for the
minimum measurable frequency shift in a noisy system for sensing applications involving
oscillators.

ω (t )

Integration

φ (t )

± X0

φ (t )

sin[(ωC ± ω )t ]

σ A (τ A )
Frequency Counting Measurement

Sω (ω )

ω2

Sφ (ω )

Ltotal (ω )

Power Spectra Measurement

Figure 2.4. Summary of the relation between different quantities. In time domain, the

phase variation φ(t), which is the integration of angular frequency variation ω (t ) ,
generates the sidebands ± x0 (φ (t ) / 2) sin[(ω c ± ω )t ] through phase modulation (PM).
The Allan deviation can be calculated with the frequency data from the frequency
counting measurements. In the frequency domain, the frequency noise density S ω (ω ) is
related to the phase noise density S φ (ω ) by S φ (ω ) = 1 / ω 2 S ω (ω ) . The noise sideband of
the carrier is characterized by Ltotal (ω ) , which can be obtained from the power spectrum
measurement.

References

A. N. Cleland and M. L. Roukes Noise processes in nanomechanical resonators.
J. Appl. Phys. 92, 2758 (2002).

A. Hajimiri and T. H. Lee The design of low noise oscillator (Norwell, Kluwer
Academic Publisher, 1999).

D. W. Allan Statistics of atomic frequency standard. Proc. IEEE 54, 221 (1966).

D. B Leeson A simple model of feedback oscillator noise spectrum. Proc. IEEE
54, 329 (1996).

Chapter 3
Theory of Phase Noise Mechanism of
NEMS

We present the theory of phase noise mechanism of NEMS. We examine both
fundamental and nonfundamental noise processes to obtain expressons for phase
noise density, Allan deviation, and mass sensitivity. Fundamental noise processes
considered here include thermomechanical noise, momentum exchange noise,
adsorption-desorption noise, diffusion noise, and temperature fluctuation noise. For
nonfundamental noise processes, we develop a formalism to consider the NyquistJohnson noise from transducer amplifier implementations. The detailed analysis
here not only reveals the achievable frequency stability of NEMS devices, but also
provides a theoretical framework to fully optimize noise performance and the mass
sensitivity for sensing applications.

3.1 Introduction
So far we have considered how physical fluctuations convert into the noise
sidebands of the carrier and give the conventional definition of phase noise, frequency
noise, and Allan deviation, all commonly used to characterize the frequency stability of
an oscillator. Here we proceed to investigate phase noise mechanisms affecting NEMS
devices. First, we examine the fundamental noise processes intrinsic to NEMS devices.1-3
We begin our discussion from thermomechanical noise, originating from thermally
driven random motion of the resonator, by considering the thermal fluctuating force
acting on the resonator. We then consider momentum exchange noise, adsorptiondesorption noise, and diffusion noise, all arising from gaseous molecules in resonator
surroundings. The impinging gaseous molecules can impart momentum randomly to a
NEMS device and induce momentum exchange noise. Moreover, when gaseous species
adsorb on a NEMS device, typically from the surrounding environment, they can diffuse
along the surface in and out of the device and produce diffusion noise. Meanwhile, they
can also briefly reside on the surface and then desorb again and generate adsorptiondesorption noise. We also discuss the noise due to the temperature fluctuations; these
fluctuations are fundamental to any object with finite thermal conductance and are
distinct from environmental drifts that can be controlled using oven-heated packaging,
similar to that used for high precision quartz clocks.
Note that the thermomechanical noise from the internal loss mechanism in the
resonator and the momentum exchange noise from gaseous damping are dissipationinduced fluctuations. They are expected for mechanical resonators with nonzero
dissipation according to the fluctuation-dissipation theorem.4 Other noise sources

including adsorption-desorption noise, diffusion noise, and temperature fluctuation noise
are parametric noise. These have to do with parametric changes in the physical properties
of the resonator such as device mass and temperature, which cause the natural resonance
frequency of the resonator to change, but do not necessarily involve energy dissipation,
leaving the quality factor unchanged.1
Finally, we consider the nonfundamental noise processes from the readout
circuitry of transducer implementations.5 In general, the NEMS transducers covert
mechanical displacement into an electrical signal, which is subsequently amplified to the
desired level by an amplifier for readout. Hence both the transducer and amplifier can
add extrinsic noise to the NEMS devices, and the impact on frequency fluctuations is
treated by our formalism developed here. Our formalism will reveal the resulting impact
on the frequency fluctuations and enable the optimization of noise performance.
Although we focus our discussion on the Nyquist-Johnson noise from the transducer and
readout amplifier implementations, it can be readily generalized to incorporate other
types of extrinsic noise such as flicker noise.
In conjunction with the discussion of each noise process, we also give the
expression for the corresponding mass sensitivity limit. In general, resonant mass sensing
is performed by carefully determining the resonance frequency ω 0 of the resonator and
then, by looking for a frequency shift in the steady state due to the accreted mass.
Therefore, the minimum measurable frequency shift, δω0 , will translate into the
minimum measurable mass, δM , referred to as the mass sensitivity, δM . Henceforth, we
model the resonator as a one-dimensional simple harmonic oscillator characterized by the

effective mass M eff and the dynamic stiffness κ eff = M eff ω 0 .6 Assuming that δM is a

small fraction of M eff , we can write a linearized expression

δM ≈

∂M eff
∂ω 0

δω 0 = ℜ −1δω 0 .

(3.1)

This expression assumes that the modal quality factor and compliance are not
appreciably affected by the accreted species. This is consistent with the aforementioned
presumption that δM << M eff . Apparently, δM critically depends on the minimum
measurable frequency shift δω 0 and the inverse mass responsivity ℜ −1 . Since κ eff for
the employed resonant mode—a function of the resonator’s elastic properties and
geometry—is unaffected by small mass changes, we can further determine that

ℜ=

ω0
∂ω 0
=−
∂M eff
2M eff

δM ≈ −2

M eff

ω0

δω 0 .

(3.2)

(3.3)

We note that equation (3.3) is analogous to the Sauerbrey equation,7 but is instead
here written in terms of the absolute mass, rather than the mass density, of the accreted
species. Both fundamental and nonfundamental noise processes will impose limits on

δω0 , and therefore on δM . For each noise process, we will integrate phase noise density
to obtain the expression for δω 0 by using equation (2.25) and translate it into δM using
equation (3.3).3

3.2 Thermomechanical Noise
We now consider the thermomechanical noise, originating from thermally driven
random motion of NEMS devices.1-3 For the one-dimensional simple harmonic oscillator,
the mean square displacement fluctuations of the center of mass,

M eff ω 0 xth

xth , satisfy

/ 2 = k B T / 2 . Here, k B is Boltzmann’s constant and T is the resonator

temperature. The spectral density of these random displacements, S x (ω ) , (with units of
m2/Hz) is given by

S x (ω ) =

S (ω )

2 2

M eff (ω − ω 0 ) + ω 2ω 0 / Q 2

(3.4)

The thermomechanical force spectral density in units of N2/Hz has a white
spectrum S F (ω ) = 4 M eff ω 0 k B T / Q . For ω >> ω 0 / Q , the phase noise density is given by
the expression1

S φ (ω ) =

1 S x (ω ) k B T ⎛ ω 0 ⎞
⎜ ⎟ .
2 xC 2
8πPC ⎝ ω ⎠

(3.5)

PC is the maximum carrier power, limited by onset of non-linearity of mechanical
vibration of the NEMS. For a doubly clamped beam with rectangular cross section driven
into flexural resonance, the non-linearity results from Duffing instability and the

maximum carrier power can be estimated by PC = ω 0 EC / Q = M eff ω 03 xC / Q with
critical amplitude xC given by t / Q(1 − ν 2 ) for doubly clamped beams.8 t is the
dimension of the beam in the direction of transverse vibration; ν is the Poisson ratio of
the beam material.9
Upon direct integration of the spectral density, Allan deviation is given by
33

σ A (τ A ) =

k BT
8PC Q 2 τ A

(3.6)

We can rewrite this expression in terms of the ratio of the maximum drive

(carrier) energy, EC = M eff ω 0 xC , to the thermal energy, Eth = k BT , representing the

effective dynamic range intrinsic to the device itself. This is the signal-to-noise ratio
(SNR) available for resolving the coherent oscillatory response above the thermal
displacement fluctuations. We can express this dynamic range, as is customary, by

DR (dB) = 10 log( EC / k B T ) in units of decibels. This yields a very simple expression

σ A (τ A ) = (1 / τ A Qω 0 )1 / 2 10 − DR / 20 .

(3.7)

We now turn to the evaluation of the minimum measurable frequency shift, δω0 ,
limited by thermomechanical fluctuations of a NEMS resonator. To obtain δω0 , the
integral in equation (2.25) must be evaluated using the expression for Sω (ω ) given in
equation (3.5) over the effective measurement bandwidth. Performing this integration for
the case where Q>>1 and 2πΔf << ω 0 / Q , we obtain:

⎡ k T ω 0 Δf ⎤
δω0 ≈ ⎢ B
⎣ EC Q ⎦

1/ 2

(3.8)
1/ 2

⎛ E ⎞ ⎛ Δf ⎞
⎟⎟ .
(3.9)
δM ≈ 2M eff ⎜⎜ th ⎟⎟ ⎜⎜
⎝ Ec ⎠ ⎝ Qω 0 ⎠
We can also recast equation (3.9) in terms of dynamic range DR and mass responsivity
ℜ as
1/ 2

1⎛ ω ⎞
δM ≈ ⎜⎜ Δf 0 ⎟⎟ 10 (− DR / 20 ) .
ℜ⎝
Q⎠

(3.10)

Note that Q / ω 0 is the open-loop response (ring-down) time of the resonator. In table 3.1,
we have translated these analytical results from equation (3.7) and equation (3.9) into

concrete numerical estimates for representative realizable device configurations. We list
the Allan deviation σ A (for averaging time τ A =1 sec) and the mass sensitivity δM (for
measurement bandwidth Δf =1 kHz), limited by thermomechancial noise, for three
representative device configurations with quality factor Q=104. For the calculation of
resonant frequency, we assume Young’s modulus E =169 GPa and mass density ρ =2.33
g/cm3 for the silicon beam and silicon nanowire and E = 1 TPa and ρ = 1g/cm3 for the
single walled nanotube (SWNT). First, a large dynamic range is always desirable for
obtaining frequency stability in the case of thermomechanical noise. Clearly, as the
device sizes are scaled downward while maintaining high resonance frequencies, M eff
and κ eff must shrink in direct proportion. Devices with small stiffness (high compliance)
are more susceptible to thermal fluctuations and consequently, the dynamic range
becomes reduced. Second, the values of the mass sensitivity span only the regime from a
few tenths to a few tens of Daltons. This is the mass range for a small individual
molecule or atom; hence it is clear that nanomechanical mass sensors offer unprecedented
ability to weigh individual neutral molecules or atoms and will find many interesting
applications in mass spectrometry and atomic physics.10,11

Device

Frequency

Dimensions (L × w × t)

Meff

σA (1sec) δM (1kHz)

Si beam

1 GHz

660 nm × 50 nm × 50 nm

2.8 fg

66 dB

3.2 × 10-10

7.0 Da

Si nanowire

7.7 GHz

100 nm × 10 nm × 10 nm

17 ag

47 dB

9.5 × 10-10

0.13 Da

10 GHz

56 nm × 1.2 nm(dia.)

14 dB

7.4 × 10

0.05 Da

SWNT

165 ag

-8

Table 3.1. Allan deviation and mass sensitivity limited by thermomechanical noise
for representative realizable NEMS device configurations

3.3 Momentum Exchange Noise

We now turn to a discussion of the consequences of momentum exchange in a
gaseous environment between the NEMS resonator and the gas molecules that impinge
upon it. Gerlach first investigated the effect of a rarefied gas surrounding a resonant
torsional mirror.12 Subsequently, Uhlenbeck and Goudmit calculated the spectral density
of the fluctuating force acting upon the mirror due to these random collisions.13
Following these analyses, Ekinci et al. have obtained the mass sensitivity of the NEMS
limited by momentum exchange noise.3 Here we reproduce a similar version of their
discussions. In the molecular regime at low pressure, the resonator’s equation of motion
is given by
.. ⎛
pAD ⎞ .
⎟⎟ x + M eff ω 02 x = F (t ) .
M eff x + ⎜⎜ M eff 0 +

(3.11)

The ( M eff ω 0 / Qi ) x term results from the intrinsic loss mechanism. The term ( pAD / v) x
represents the drag force due to the gas molecules. P is the pressure, AD is the device
surface area, and v = k B T / m is the thermal velocity of gas molecule. The quality factor
due to gas dissipation can be defined as Q gas = MvPAD . The loaded quality factor QL , as
−1
a result of two dissipation mechanisms, can be defined as Q L−1 = Qi−1 + Q gas
. Since we

have treated the thermomechanical noise from the intrinsic loss mechanism, we assume
that Qi >> Q gas and focus on the noise from gaseous damping. The collision of gas
molecules produces a random fluctuating force with the spectral density given by3
S F (ω ) = 4mvPAD =

4 Mω 0 k B T
Q gas

(3.12)

Similar to equation (3.5) and equation (3.6), the resulting formulas for the phase
noise density and the Allan deviation are

⎛ ω0 ⎞
S φ (ω ) =
⎜ ⎟ ,
8πPC Q gas ⎝ ω ⎠
k BT

σA =

(3.13)

k BT 1
PC Q gas
τA

(3.14)
After taking similar steps leading to equation (3.9), we obtain
⎛E ⎞
δM ≈ 2M eff ⎜⎜ th ⎟⎟
⎝ Ec ⎠

1/ 2

⎛ Δf ⎞
⎜Q ω ⎟
⎝ gas 0 ⎠

1/ 2

(3.15)

3.4 Adsorption-Desorption Noise

Adsorption-desorption noise has been first discussed by Yong and Vig.14 The
resonator environment will always include a nonzero pressure of surface contaminated
molecules. As the gas molecules adsorb and desorb on the resonator surface, they mass
load the device randomly and cause the resonant frequency to fluctuate. Yong and Vig
developed the model for noninteracting, completely localized monolayer adsorption,
henceforth referred to as Yong and Vig’s model. In addition to Yong and Vig’s model,
we present the ideal gas model for the case of noninteracting, completely delocalized
adsorption. However, the extreme of completely localized or completely delocalized
adsorption rarely occurs on real surfaces; the adsorption on real surfaces always lies
between these two extremes.15 Adsorbed gases molecules can interact with each other,
resulting in phase transitions on the surface.16 Instead of monolayer adsorption, multilayer
38

adsorption usually happens on real surfaces.15 All these effects can further complicate the
analysis of adsorption-desorption noise. The two models presented here, despite their
simplicity, reveal valuable insight in the theoretical understanding of the adsorptiondesorption noise.
In Yong and Vig’s model, the assumption of localized adsorption means that the
kinetic energy of the adsorbed molecule is much smaller than the depth of surface
potential, and thus the adsorbed molecule is completely immobile in the later direction.
Thus the concept of adsorption site on the surface is well defined. We further assume
each site can accommodate only one molecule and consider the stochastic process of
adsorption-desorption of each site. Consider a NEMS device surrounded by the gas with
pressure, P, and temperature, T. From kinetic theory of gas, the adsorption rate of each
site is given by the number of impinging atoms or molecules per unit time per unit area
times the sticking coefficient, s, and the area per site Asite.
ra =

2 P
sAsite ,
5 mkT

(3.16)

where P and T are the pressure and temperature of gas, respectively. In general, the
sticking coefficient depends on temperature and gaseous species.17 Here we assume that
the sticking coefficient is independent of the temperature.
Once bound to the surface, a molecule desorbs at a rate
rd = ν d exp(−

Eb
),
kT

(3.17)

ν d is the desorption attempt frequency, typically of order 1013 Hz for a noble gas on a
metallic surface, and Eb is the binding energy. For N molecules adsorbed on the surface,
the total desorption rate for the whole device is Nrd. Since each site can only

accommodate one molecule, the number of available sites for adsorption is Na-N, so the
total adsorption rate is (Na-N)ra. Equating these two rates, we obtain the number of
adsorbed molecules
N = Na

ra
ra + rd

(3.18)

The average occupation probability f of a site is defined as the ratio of the
adsorbed molecules to the total number of sites, N/Na, and is given by f = ra /(ra + rd ) .
Substitution of equation (3.16) and equation (3.17) into equation (3.18) yields the
formula for the number of adsorbed molecules as a function of temperature, also known
as the Langmuir adsorption isotherm.16
2 p
Asite exp( b )
5 mkT ν d
kT
a(T ) p ,
Na 2 p
1 + a(T ) p
Asite exp( b ) + 1
5 mkT ν d
kT

a (T ) =

2 P
exp( b ) .
5 mkT ν d
kT

(3.19)

(3.20)

We can rewrite equation (3.16) in terms of the gaseous flux, Φ flux , given by

Φ flux = (2 / 5)( P / mkT ) .
Eb
νd
kT .
Na
Φ flux
Asite exp( b ) + 1
νd
kT

Φ flux

Asite exp(

(3.21)

We derive the spectral density of the frequency noise by considering the
stochastic process of the adsorption-desorption of each site, which can be described by a
continuous time two state Markov chain.14 Here we briefly sketch the derivation for a two
state Markov chain.18 Since each site can be occupied or unoccupied, we consider a
40

continuous time stochastic process { ζ (t ) , t>0}, where the random variable ζ( t ) can take
either 0 (unoccupied) or 1 (occupied). The two rate constants of such a Markov chain are
rd, the rate from state 1 (occupied state) to state 0 (unoccupied), and ra, the rate from state
0 (unoccupied state) to state 1 (occupied state).We define Pij (t ) as the conditional
probability that a Markov chain, presently in state i, will be in the state j after additional
time t. Assuming that the site is initially occupied, we have initial condition, P11 (0) = 1 ,
and for a two state system, P10 (t ) = 1 − P11 (t ) . The corresponding Kolmogorov’s forward
equation and its solution are given by10
dP11
= rd P10 (t ) − ra P11 (t ) ,
dt
P11 (t ) =

(3.22)

ra
rd
e −( ra + rd ) t = f + (1 − f )e −t / τr .
ra + rd ra + rd

(3.23)

The correlation time τ r is defined as 1 /(ra + rd ) . The autocorrelation function can
be found by calculating the expectation value of ζ (t + τ )ζ (t ) from the conditional
probability function. By definition, the autocorrelation function of ζ (t ) is given by
Rsite (τ ) = E[ζ (t + τ )ζ (t )] = σ OCC

− τ /τ r

+ f.

(3.24)

E[] denotes the expectation value of the random variable. Here for our purpose,
we neglect the constant term f since this corresponds to the D.C. part of the spectra. σ OCC
is the variance of occupational probability f, given by σ OCC
= f (1 − f ) = ra rd /(ra + rd ) 2 .
Note that σ OCC
reaches a maximum for f=0.5 when the adsorption and desorption rates of

the site are equal.

We apply the Wiener-Khintchine theorem to obtain the corresponding spectral
density of ζ (t ) for each site by performing the Fourier transform of equation (3.24).
S ζ (ω ) =

2σ OCC
τr /π
1 + (ωτ r ) 2

(3.25)

Each adsorbed molecule of mass m will contribute to fractional frequency change
m/2Meff. We obtain the spectral density of fractional frequency noise by simply summing
the contribution from each individual site.

2σ OCC
N a / π ⎛⎜ m ⎞⎟
S y (ω ) =
1 + (ωτ r ) 2 ⎜⎝ 2M eff ⎟⎠

(3.26)

Since the spectral density exhibits Lorentizian function form, we use equation (2.16) to
obtain

σ A (τ A ) = N a σ OCC

M eff

τA

F( r ) .

(3.27)

F ( x) is the analytic function defined in equation (2.17). In the limit, τ r << τ A , equation
(3.27) becomes

σ A (τ A ) = N a σ OCC

M eff

τr
2τ A

(3.28)

In the other limit, τ A << τ r , equation (3.27) becomes

σ A (τ A ) = N a σ OCC

M eff

τA
6τ r

(3.29)

In the ideal gas model, the assumption of delocalized adsorption means that the
kinetic energy of the adsorbed molecule is much higher than the depth of the surface
potential, and thus the adsorbed molecule is mobile in the lateral direction. The notion of
adsorption site in Yong and Vig’s model is not well defined.14 We thus analyze the
42

kinetics of adsorption-desorption using the total adsorption and desorption rates of the
adsorbed atoms on the device. The total adsorption rate of the device is given by the flux
of molecules multiplied by the sticking coefficient s and the device area AD ,
Ra =

sAD .
5 mk B T

(3.30)

Once bound to the surface, the molecule desorbs at a rate given by
rd = ν d exp(− Eb / kT ) . The total desorption rate of all the adsorbed molecules on the

device is simply
Rd = ν d exp(−

Eb
)N .
kT

(3.31)

At equilibrium, the total adsorption rate equals the total desorption rate, and the
number of adsorbed molecules is given by
2 s
AD 5 ν d
b(T ) =

2 s
5νd

mkT
mkT

exp(

Eb
) = b(T ) P ,
kT

(3.32)

Eb
).
kT

(3.33)

exp(

We also rewrite the expression in terms of the impinging gaseous flux Φ flux ,
= Φ flux exp( b ) .
AD ν d
kT

(3.34)

We derive the spectral density of the fractional frequency noise by considering the
dilute gas limit of Yong and Vig’s model. This is done by keeping the number of
adsorbed molecules, N = fN a , constant, and letting the occupational probability go to
zero, and N a go to infinity. Hence, σ OCC
N a = f (1 − f ) N a → N . The spectral density of

fractional frequency noise becomes
43

2 N / π ⎛⎜ m ⎞⎟
S y (ω ) =
1 + (ωτ r ) 2 ⎜⎝ 2M eff ⎟⎠

(3.35)

The correlation time due to adsorption-desorption cycle is given by the time constant of
the rate equation
dN
= Ra − Rd = Ra − ν d exp(− b ) N .
kT
dt

(3.36)

We find that

τ r = ν d exp(

Eb
).
kT

(3.37)

Since the spectral density of fractional frequency in equation (3.35) exhibits
Lorentizian function form, we use equation (2.16) to obtain

σ A (τ A ) = N

M eff

τA

F( r ) .

(3.38)

In the limit, τ r << τ A , this expression becomes

σ A (τ A ) = N

M eff

τr
2τ A

(3.39)

In the other limit, τ A << τ r , this expression becomes

σ A (τ A ) = N

M eff

τA
6τ r

(3.40)

Table 3.3 tabulates the expressions for the two models presented here. Note that equation
(3.27) differs from equation (3.38) in the statistics. The occupational variance σ OCC
in

equation (3.27) and thus adsorption-desorption noise in Yong and Vig’s model vanishes
upon completion of one monolayer due to the assumption that each site accommodates

only one molecule. In contrast, equation (3.38) exhibits idea gas statistics, manifested in
the square root dependence of the number of adsorbed molecules.
Now we discuss the effect of the correlation time on Allan deviation. Because the
spectral density of fractional frequency for these two models exhibits Lorentizian
functional form, both equation (3.27) and equation (3.38) have the same dependence on
the ratio of the correlation time, τ r , to the averaging time, τ A , through the analytic
function, F ( x) , defined in equation (2.17). Mathematically, F ( x) reaches a maximum at
0.095 for x=1.85 and vanishes when x equals to zero or infinity, and. In other words, the
adsorption-desorption noise in both models maximizes when τ r = 0.095τ A and
diminishes for τ r >> τ A or τ r << τ A with the asymptotic behaviors dictated by equation
(3.25), equation (3.26), equation (3.37), and equation (3.38).
To explicitly illustrate the surface effect of adsorption-desorption noise, we give
the expression for the maximum Allan deviation σ A max in Yong and Vig’s model by
simultaneously maximizing σ OCC and F (τ r / τ A ) in equation (3.27). We find that

σ A max = 0.3

⎛ m ⎞ Na
⎛ m ⎞ 1 Na
⎟⎟
⎟⎟
= 0.3⎜⎜
= 0.3⎜⎜
N a mD
⎝ mD ⎠ NV
⎝ mD ⎠ NV NV
Na m

(3.41)

Here m D is the mass of a single atom adsorbed the device. N V is the total number of
atoms of the device. N a / N V is the surface-to-volume ratio.
Finally, we give the expressions for minimum measurable frequency shift and
mass sensitivity. For Yong and Vig’s model, the integration of the spectra density yields

δω0 =

1 mω0σ occ
[N a arctan(2πΔfτ r )]1 / 2 ,
2π M eff

(3.42)
45

δM ≈

1/ 2
mσ occ [N a arctan(2πΔfτ r )] .
2π

(3.43)

Similar to equation (3.41), we give the expression for the maximum mass fluctuation

δM max by the maximized σ OCC and arctan(2πΔfτ r ) in equation (3.43) from Yong and
Vig’s model. We find that δM max ≈1/ 32π N a m when 2πΔfτ r → ∞ and f=0.5.
Similarly, for ideal gas model, we obtain

δω 0 =

1 mω 0
[N arctan(2πΔfτ r )]1 / 2 ,
2π M eff

(3.45)

δM ≈

1/ 2
m[N arctan(2πΔfτ r )] .
2π

(3.46)

Table 3.2 summarizes the expressions from Yong and Vig’s and ideal gas models.
Table 3.3 shows the numerical estimates of σ A max and δM max arising from nitrogen for
the same representative NEMS devices used in table 3.1. (The number of sites, N a , is
calculated assuming each atom on the device surface serves as one adsorption site. For
silicon beam and nanowire, we assume that the device surface is terminated Si(100) with
lattice constant=5.43 Å. For a single-walled nanotube (SWNT), we assume that the
carbon bond length is 1.4 Å.) First, the magnitude of δM max indicates that the mass
fluctuation associate with adsorption-desorption noise of NEMS is at zeptogram level.
Second, table 3.4 shows the increase of Allan deviation as a result of increasing the
surface-to-volume ratio as the device dimensions are progressively scaled down. In
particular, for the 10 GHz single-walled nanotube (SWNT), representing the extreme
case that all the atoms are on the surface, the corresponding Allan deviation is almost five
orders of magnitude higher than that due to thermomechanical noise (see table 3.1). In
other words, the adsorption-desorption noise can severely degrade the noise performance
46

of the device. This, however, can be circumvented by packaging the device at low
pressure or passivating the device surface.

Table 3.2. Summary of Yong and Vig’s and ideal gas models

Adsorption

Yong and Vig

Ideal Gas

Localized

Delocalized

ra =

Rates

2 p
sASite
5 mkT

Ra =

rd = ν d exp( E b / k B T )

R d = ν d exp( E b / k B T ) N

a(T ) p
N a 1 + a(T ) p

Isotherm
a(T ) =

Correlation Time

Spectral Density

= b(T ) p
AD

exp( b ) Asite b(T ) = 2
exp( b )
5 mkTν d
kT
5 mkTν d
kT

τ r = 1 /(ra + rd )

τ r = 1 /ν d exp( Eb / kT )

2 N aσ OCC
τ r / π ⎛⎜ m ⎞⎟
S y (ω ) =
2 2
⎜M ⎟
1+ ω τr
⎝ eff ⎠

σ A = σ OCC N a

Allan deviation

sAD
5 mk B T

M eff

2 Nτ r / π ⎛⎜ m ⎞⎟
S y (ω ) =
1 + ω 2τ r2 ⎜⎝ M eff ⎟⎠

τr
2τ A

σA = N

M eff

τr
2τ A

σ OCC
= ra rd /(ra + rd ) 2

Device

Frequency

Na/NV

σAmax(gas)

δMmax

Si beam

1 GHz

1.1 × 10-2

8.9 × 105

1.7 × 10-6

1.6 zg

Si nanowire

7.7 GHz

5.5 × 10

2.7 × 10

4.9 × 10

0.28 zg

SWNT

10 GHz

5.0 × 103

4.9 × 10-3

0.27 zg

-2

-5

Table 3.3. Maximum Allan deviation and mass fluctuation of representative NEMS
devices

3.5 Diffusion Noise

So far we have analyzed the adsorption-desorption noise from adsorbed gasous
species on the NEMS device. The surface diffusion provides another channel for
exchange of adsorbed species between the device and the surroundings to generate noise.
We start the analysis of diffusion noise from calculating the autocorrelation function of
fractional frequency fluctuation. Mathematically, the autocorrelation function G (τ ) is
calculated as the time average ( <> ) of the product of the frequency fluctuations of the
NEMS.
G (τ ) =< δf (t )δf (t + τ ) > / < f (t ) > 2 =< ∫ δf ( x, t )dx ∫ δf ( x' , t + τ )dx' > / < f (t ) > 2 .

(3.47)

Here f (t ) is the instantaneous resonant frequency of the device and we define the
averaged resonant frequency by < f (t ) >≡ f 0 . In the actual experiments, δf ( x, t ) remains
proportional to local concentration fluctuation δC ( x, t )dx and is given by

δf ( x, t )
f0

=−

m u ( x) 2 δC ( x, t )dx
2M eff 1
u ( x) dx
L∫

(3.48)

where m is the mass of the adsorbed atoms or molecules, M eff is the effective vibratory
mass of the device,3 L is the length of the device, and u ( x) is the eigenfunction
describing flexural displacement of the beam. Here we only consider the fundamental
mode u ( x) = 0.883 cos kx + 0.117 cosh kx for a beam extending from − L / 2 to L / 2 , with
kL = 4.730 with doubly clamped boundary condition imposed. Note that the end of the

beam is never perfectly clamped so doubly clamped boundary condition is only an
approximation. The normalization of u(x) factors out in equation (3.48); therefore we are

free

choose

u(0)=1.

define

Green

function

for

diffusion

φ ( x, x' ,τ ) =< δC ( x, t + τ )δC ( x' , t ) > . As a result, equation (3.48) becomes
L/2

G (τ ) = L2 (

L/2

∫ dx ∫ dx'u ( x) u ( x' ) φ ( x, x' , t ) >

m 2
m 2 −L / 2 −L / 2
) = L2 (
2M eff
2M eff
⎡ L/2
⎢ ∫ u ( x) dx ⎥
⎣− L / 2

(3.49)

In case of pure diffusion of one species in one dimension, the concentration

δC ( x,τ ) obeys the diffusion equation
∂δC ( x,τ )
∂ 2δC ( x,τ )
=D
∂τ
∂2x

(3.50)

Following Elson and Magde,19,20 we find that

φ ( x, x ' , τ ) =

4πDτ

exp[−

( x − x' ) 2
],
4 Dτ

(3.51)

where N is the average total number of the adsorbed atoms inside the device. To calculate
the autocorrelation function, we can approximate the vibrational mode shape u ( x) by a
1 ax
Gaussian mode shape exp[− ( ) 2 ] with a numerical factor a=4.43, extending from -∞
2 L
to ∞. Figure 3.1 shows the true vibration mode shape of the beam with its Gaussian
approximation. Using Gausssian approximation, we can perform the integral analytically
and obtain the autocorrelation function of the fractional frequency noise
L/2
L/2
⎡ L/2
m 2
G (τ ) = L (
) ∫ dx' ∫ dx[u ( x) u ( x' ) φ ( x, x' ,τ ) / ⎢ ∫ u ( x" ) 2 dx"⎥
2 M eff − L / 2 − L / 2
⎣− L / 2

⎡∞
m 2
≈L(
) ∫ dx ∫ dx'[u ( x) 2 u ( x' ) 2 φ ( x, x' ,τ ) / ⎢ ∫ u ( x" ) 2 dx"⎥
2 M eff − ∞ − ∞
⎣− ∞

(3.52)

aN
m 2
1/ 2
2π 2 M eff (1 + τ / τ D )

Here the diffusion time is defined by τ D = L2 /(2a 2 D ) . Note that the time course of G (τ )
is determined by the factor (1 + τ / τ D ) −1 / 2 even if the concentration correlation function
has a typical exponential time dependence. This results from the convolution of the
exponential Fourier components of diffusion with the Gaussian profile of the mode
shape.20 Also note that G (τ ) is of the form (1 + τ / τ D ) −1 / 2 d

with d = 1, the

dimensionality of the problem. This is consistent with the factor (1 + τ / τ D ) −1 / 2 d , obtained
by Elson and Magde with d = 2.
We then apply the Wiener-Khintchine theorem to obtain the corresponding
spectral density by
S y (ω ) =

iωτ
∫ G(τ )e =

π −∞

2aN

3/ 2

cos ωτ
m 2
) ∫
dτ
2 M eff 0 (1 + τ / τ D )1 / 2

aN m 2
) τ Dξ (ωτ D ).
4π M eff

(3.53)

Here ξ ( x) ≡ (cos( x) + sin( x) − 2C ( x ) cos( x) − 2 S ( x ) sin( x)) / x and C (x) and S (x)
are Fresnel integrals defined by21
C ( x) =

cos u du ,
π∫

(3.54)

S ( x) =

sin u du .
π∫

(3.55)

In figure 3.2, we plot the function ξ ( x) with its asymptotic forms: ξ ( x) = 1 / x as
x → 0 and ξ (x) = 1 / x 2 2π as x → ∞ . For ω << 1 / τ D , the spectral density of

fractional frequency noise is given by
S y (ω ) =

aN m 2
) τD
4π M eff
ωτ D

(3.56)
52

For ω >> 1 / τ D , the spectral density of fractional frequency noise is given by
S y (ω ) =

4 2π

3/ 2

m 2 1
M eff ω 2τ D

(3.57)

We now obtain the expression for Allan deviation using equation (3.53) by the
performing the following integration,

2aN ⎛⎜ m ⎞⎟
σ (τ A ) = ∫
ωτ
Χ ( D ).
sin
π ⎝ M eff ⎠
τA
0 (ωτ A )

(3.58)

Here Χ ( x) is defined as

Χ ( x) = x ∫ ξ (ηx)

sin 4 (η / 2)

η2

dη .

(3.59)

For x → ∞ , the asymptotic form of Χ (x) is given by

Χ ( x) =

π 1/ 2 1
24 2 x

(3.60)

In figure 3.3, we plot the function Χ (x) in equation (3.60) together with its asymptotic
form. For the limit, τ D >> τ A , we give the expression for Allan deviation as1

aN ⎛⎜ m ⎞⎟ τ A
σ A2 (τ A ) = ∫
sin
ωτ
2 y
12π ⎜⎝ M eff ⎟⎠ τ D
0 (ωτ A )

(3.61)

1.0

u(x)

0.8
0.6
0.4
0.2
0.0

-0.4

-0.2

0.0

0.2

0.4

x/L
Figure 3.1. Vibrational mode shape of the beam with doubly clamped boundary
condition imposed and its Gaussian approximation. The vibrational beam mode shape

(black) with doubly clamped boundary condition imposed is displayed with its Gaussian
approximation (red).

ξ(x)

-2

-4

10
0.01

0.1

100

Figure 3.2. Plot of the function ξ ( x ) . The function ξ ( x) (black solid) is plotted

together with it asymptotic approximations 1 / x (red dash) as x → 0 and
1 / x 2 2π (blue dash) as x → ∞ .

-1

-2

-3

-4

Χ(x)

-2

-1

Figure 3.3. Plot of Χ ( x ) and its asymptotic form. The function Χ ( x) (black solid) is

plotted together with its asymptotic form (red dash) Χ ( x) =

π 1/ 2 1
24 2 x

as x → ∞ .

3.6 Temperature Fluctuation Noise

The small dimensions of NEMS resonators in general imply that the heat capacity
is very small and therefore the corresponding temperature fluctuations can be rather
large. The effect of such fluctuations depends on upon the thermal contact of the NEMS
to their environment. Because the resonant frequency depends on the temperature through
the resonator material parameters and geometric dimensions, the temperature fluctuations
produce frequency fluctuations. Here we present a simple model using the thermal circuit
consisting of a heat capacitance, c , connected by a thermal conductance, g , to an infinite
thermal reservoir at temperature, T.

In the absence of any power load, the heat

capacitance, c , will have an average thermal energy, EC = cT . Changes in temperature
relax with thermal time constant, τ T = c / g . Applying the fluctuation-dissipation theorem
to such a circuit, we expect a power noise source, p , connected to the thermal
conductance, g , with the spectral density, S p (ω ) = 2k B T 2 g / π , and cause the
instantaneous energy, E C (t ) = E C + δE (t ) , to fluctuate.4 The spectral density of the
energy fluctuations δE (t ) can be derived as

S E (ω ) =

2 k BT 2 c 2 / g
π 1 + ω 2τ T2

(3.62)

We can interpret the energy fluctuations as temperature fluctuations δTC (t ) , if we
define the temperature as TC = EC / c . The corresponding spectral density of the
temperature fluctuations is given by

S T (ω ) =

2 k BT / g
π 1 + ω 2τ T2

(3.63)

Equation (3.63) applies to any system that can be modeled as a heat capacitance
with a thermal conductance. For a doubly clamped beam, however, there is no clear
separation of the structure into a distinct heat capacitance and a thermal conductance.
Cleland and Roukes have developed a distributed model of thermal transport along a
doubly clamped beam of constant cross section, and derived the spectral density of
frequency fluctuations arising from temperature fluctuations of a NEMS resonator.1 Their
analysis leads to

S T (ω ) =

4 k BT 2 / g
π 1 + ω 2τ T2

(3.64)

⎛ 22.4
2 ∂c s ⎞ 1 k B T 2 / g
S y (ω ) = ⎜⎜ − 2 2 α T +
⎟ π 1 + ω 2τ 2 .

Here

cs = E / ρ

the

temperature

(3.65)

dependent

speed

sound,

α T = (1 / L)∂L / ∂T is the linear thermal expansion coefficient, and g and τ T are the
thermal conductance and thermal time constant for the slice, respectively. In the limits

τ A >> τ T , the Allan deviation is given by

σ A (τ A ) =

2k B T 2 1 ⎛ 22.4
2 ∂c s ⎞
⎟ .
⎜ − 2 2 αT +
gτ A τ T ⎝ ω 0 L
c s ∂T ⎟⎠

(3.66)

To give the expression for δω 0 and δM , we integrate equation (3.65) over the
measurement bandwidth and obtain
⎡ 1 ⎛ 22.4c 2
∂c s ⎞ ω 0 k B T 2 arctan(2πΔfτ T ) ⎤
δω 0 = ⎢ 2 ⎜⎜ −
τT
c s ∂T ⎟⎠
⎢ 2π ⎝ ω 0 2 l 2

1/ 2

(3.67)

⎛ 22.4c s 2
2 ∂c s ⎞⎟ ⎡ k B T 2 arctan(2πΔfτ T ) ⎤
δM = 1 / 2 2M eff ⎜ −
αT +
2 2
c s ∂T ⎟⎠ ⎣
gτ T
⎝ ω0 l

1/ 2

(3.68)

The values of the material dependent constants for silicon have been calculated
as1

⎛ 22.4
2 ∂c s ⎞
⎜ − 2 2 αT +
⎟ = 1.26 × 10-4 1/K.
⎜ ω L
c s ∂T ⎟⎠

(3.69)

g = 7.4× 10-6 W/K and τ T = 30 ps. Using these values, a numerical estimate of equation
(3.66) for 1 GHz silicon beam in table 3.1 is given by σ A (τ A ) = 9.3x10-11/ τ A .1 For

τ A = 1 sec, the Allan deviation is 9.3x10-11, of the same order of magnitude as that due to
the thermomechanical noise at room temperature listed in table 3.1. Similarly, for the
same device at room temperature with measurement bandwidth Δf = 1 Hz, we obtain

δM = 0.245 Da. Despite of the role of thermal fluctuations in generating phase noise that
limits the mass sensitivity, single Dalton sensing is readily achievable. The effect can be
even more significant as we further scale down the dimensions or increase the device
temperature. This can be circumvented by lowering the temperature or optimizing the
thermal contact of the NEMS to its environment.

3.7 Nonfundamental Noise

We develop a simple formalism to consider nonfundamental noise process from
transducer amplifier implementations of NEMS.5 First, the spectral density of the
frequency noise S ω (ω ) is transformed into the voltage domain by the displacement
59

transducer, the total effective voltage noise spectral density at the transducer’s output
predominantly originates from the transducer and readout amplifier.5 It is the total
voltage noise referred back to the frequency domain that determines the effective
frequency fluctuation spectral density for the system S ω (ω ) = SV /(∂V / ∂ω ) 2 . V is the
transducer output voltage. If we define the transducer responsivity by the derivative of
transducer output voltage with respect to displacement, i.e., RT = (∂V / ∂x) , a simple
estimate is given by (∂V / ∂ω ) ≈ QRT xC / ω 0 . Assuming the voltage fluctuation SV
results from Nyquist-Johnson noise from the transducer amplifier and thus has a white
spectrum, using equation (2.15) we obtain the expression for the Allan deviation

σ A (τ A ) =

1 (πSV / τ A )1 / 2
RT xC

(3.70)

We can rewrite this equation in a simple form in terms of the dynamic range,

DR = 20 log[ RT2 xC /(πSV / τ A )1 / 2 ] , or equivalently the signal-to-noise ratio (SNR)
referred to transducer output of the NEMS.

σ A (τ A ) =

1 - DR / 20
10

(3.71)

Finally, we give the expression for the minimum detectable frequency shift δω
and mass sensitivity δM . Upon the integration of spectral density using equation (2.21),
the minimum detectable frequency shift for the measurement bandwidth Δf , is simply

δω =

ω 0 ( SV Δf )1 / 2

RT xC

ω0

10 - DR / 20 .

(3.72)

The mass sensitivity follows as

δM ~ 2( M eff / Q)10 − DR / 20 .

(3.73)
60

Equation (3.73) indicates the essential considerations for optimizing NEMS based mass
sensors limited by the Nyquist-Johnson noise. First, this emphasizes the importance of
devices possessing low mass, i.e., small volume, while keeping high Q. Second, the
dynamic range for the measurement should be maximized. This latter consideration
certainly involves careful engineering to minimize the noise from transducer amplifier
implementations and controlling the nonlinearlity of the resonator through the mechanical
design.

3.8 Conclusion

We present the theory of phase noise mechanisms affecting NEMS. We examine
both fundamental and nonfundamental noises and their imposed limits on device
performance. Table 3.4 tabulates the expressions for fundamental noise processes
considered in this work. We find that the anticipated noise is predominantly from
thermomechanical noise, temperature fluctuation noise, adsorption-desorption noise, and
diffusion noise. First, a large dynamic range is always desirable for obtaining frequency
stability in the case of thermomechanical noise. Clearly, as the device sizes are scaled
downward while maintaining high resonance frequencies, M eff and κ eff must shrink in
direct proportion. Devices with small stiffness (high compliance) are more susceptible to
thermal fluctuations and consequently, the dynamic range becomes reduced. Second, next
generation NEMS appear to be more susceptible to temperature fluctuations—more
intensively at elevated temperatures. This fact can be circumvented by lowering the
device temperatures and by designing NEMS with better thermalization properties. Third,
for adsorption-desorption noise, both Yong and Vig’s and ideal gas model suggest that
61

this noise becomes significant when appreciable molecules adsorb on the NEMS surface
and the correlation time of adsorption-desorption cycle roughly matches the averaging
time. One could easily prevent this, for instance, by reducing the packaging pressure or
passivating the device to change the binding energy between the molecule and the
surface.
To evaluate the impact of each noise process on the mass sensing application, we
give expressions for the minimum measurable frequency shift and mass sensitivity. Our
analysis culminates in the expression equation (3.10), i.e.,
1/ 2

1⎛ ω ⎞
δM ≈ ⎜⎜ Δf 0 ⎟⎟ 10 (− DR / 20 ) .
ℜ⎝
Q⎠

(3.74)

Equation (3.74) distills and makes transparent the essential considerations for
optimizing inertial mass sensors at any size scale.

There are three principal

considerations. First, the mass responsivity, ℜ , should be maximized. As seen from
equation (3.3), this emphasizes the importance of devices possessing low mass, i.e., small
volume, which operate with high resonance frequencies.

Second, the measurement

bandwidth should employ the full range that is available. Third, the dynamic range for the
measurement should be maximized. At the outset, this latter consideration certainly
involves careful engineering to minimize nonfundamental noise processes from the
transducer amplifier implementation, as expressed in equation (3.72) and equation (3.73).
But this is ultimately feasible only when fundamental limits are reached. In such a
regime it is the fundamental noise processes that become predominant.
In table 3.1, we have translated the analytical results from equation (3.10) into
concrete numerical estimates for representative and realizable device configurations. The
values of δM span only the regime from a few tenths to a few tens of Daltons. This is the
62

mass range for a small individual molecule; hence it is clear that nanomechanical mass
sensors offer unprecedented sensitivity to weigh individual neutral molecules routinely—
blurring the distinction between conventional inertial mass sensing and mass
spectrometry.11

Table 3.4 Summary of expressions for spectral density and Allan deviation for
fundamental noise processes considered in this work
Noise
Thermomechanical

Correlation Time
None

Expression
Sφ (ω ) =

k BT ⎛ ω 0 ⎞
8πPc Q 2 ⎝ ω ⎠

σ A (τ A ) =

Momentum Exchange

None

Sφ (ω ) =

τ r = 1/(ra + rd )

S y (ω) =

k BT
8PC Q 2τ A

k BT ⎛ ω0 ⎞
2 ⎜
8πPcQgas ⎝ ω ⎠

σ A (τ A ) =

Adsorption-Desorption
Yong and Vig’s Model

Ideal Gas Model

Naτ r / π m 2
2σ OCC
1 + ω 2τ r
2Meff

S y (ω) =

τ D = L2 / 2a 2 D

Temperature Fluctuation

τT = c / g

σ OCC m

τA

F( r )

M eff

2Nτ r / π ⎛⎜ m ⎞⎟
1 + ω2τ r ⎜⎝ 2Meff ⎟⎠

σ A (τ A ) = N

Diffusion

k BT
8 PC Qgas
τA

σ A (τ A ) = N a

τ r = 1 /ν d exp(Eb / k BT )

M eff

τA

F( r )

σ A (τ A ) =

2aN ⎛⎜ m ⎞⎟
Χ( D)
π ⎜⎝ M eff ⎟⎠
τA

⎡ 1 ∂ω ⎤ 4 k BT / g
S y (ω ) = ⎢ ( 0 )⎥
2 2
⎣ ω0 ∂T ⎦ T π 1 + ω τ T

σ A (τ A ) =

4k BT 1 1 ⎛ ∂ω0 ⎞
gτ A τ T ω0 ⎝ ∂T ⎠

References

A. N. Cleland and M. L. Roukes Noise processes in nanomechanical resonators. J.
Appl. Phys. 92, 2758 (2002).

J. Vig and Y. Kim Noise in MEMS resonators. IEEE Trans. on Ultrasonics
Ferroelectics and Frequency Control. 46, 1558 (1999).

K. L. Ekinci, Y. T. Yang, M. L. Roukes Ultimate limits to inertial mass sensing
based upon nanoelectromechanical systems. J. Appl. Phys. 95, 2682 (2004).

L. D. Landau and E. M. Lifshitz Statistical Physics (England, Oxford, 1980).

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes Ultrasensitive
nanoelectromechanical mass detection. Appl. Phys. Lett. 84, 4469 (2004).

For the fundamental mode response of a doubly clamped beam with rectangular
cross section, the effective mass, dynamic stiffness are given as M eff = 0.735ltwρ ,

κ eff = 32 Et 3 w / L3 . Here, L, w, and t are the length, width and thickness of the
beam. E is Young’s modulus and ρ is the mass density of the beam. We have
assumed the material is isotropic; for single-crystal device anisotropy in the
elastic constants will result in a resonance frequency that depends upon specific
crystallographic orientation.
7.

G. Z. Sauerbrey Verwendung von Schwingquarzen zur Wagang dunner Schichten
und zur Mikrowagung Z. Phys. 155, 206-222 (1959).

H. A. C. Tilmans and M. Elwenspoek, and H. J. Flutiman Micro resonator force
gauge. Sens. Actuators A 30, 35 (1992).

For a doubly clamped tube of diameter d , we can calculate the maximum carrier
power using xC = d / 2 / 0.5Q (1 − ν 2 ) . See A. Husain et al. Nanowire-based
very high frequency electromechanical resonator. Appl. Phys. Lett. 83, 1240
(2003).

10.

W. Hansel, P. Hommelhoff, T. W. Hansh, and J. Reichel. Bose-Einstein
condensation on microelectronic chips. Nature 413, 498-500 (2001).

11.

R. Aebersold and M. Mann Mass spectrometry-based proteomics. Nature 422,
198-207 (2003).

12.

W. Gerlach Naturwiss. 15, 15 (1927).

13.

G. E. Uhlenbeck and S. A. Goudsmit A problem in brownian motion. Phys. Rev.
34, 145 (1929).
65

14.

Y. K. Yong and J. R. Vig Resonator surface contamination: a cause of frequency
fluctuations. IEEE Trans. on Ultrasonics Ferroelectics and Frequency Control.
36, 452 (1989).

15.

H. Clark The Theory of Adsorption and Catalysis (London, Academic Press,
1970).

16.

S. Ross and H. Clark On physical adsorption VI two dimensional critical
phenomena of xenon, methane, ethane adsorbed separately on sodium chloride. J.
Am. Chem. Soci.76, 4291 (1954).

17.

H. J. Kreuzer and Z. W. Gortel Physisorption Kinetics (Heidelberg, SpringerVerlag, 1986).

18.

S. M. Ross Stochastic Process (New York, John Wiley & Sons, 1996).

19.

E. L. Elson and D. Magde Fluoresence correlation spectroscopy I concept basis
and theory. Biopolymer 13, 1-27 (1974).

20.

D. Magde, E. L. Elson, and W. W. Webb Thermodynamic fluctuations in a
reacting system- Measurement by fluorescence correlation spectroscopy. Phys.
Rev. Lett. 29, 705-708 (1972).

21.

I. S. Gradshteyn and I. M. Ryzhik Alan Jefferey, Editor Table of Integrals,
Series, and Products 5th edition (New York, Academic Press, 1980).

Chapter 4
Experimental Measurement of Phase
Noise in NEMS

We present the experimental measurement of phase noise of NEMS. First, we
analyze control servo behavior of the phase-locked loop, and give expressions for the
locked condition and loop dynamics. We then describe two implementation schemes
at very high frequency and ultra high frequency bands: (1) homodyne detection
phase-locked loop based on a two port NEMS device and (2) frequency modulation
phase-locked loop. The achieved phase noise and Allan deviation are compared with
the local oscillator requirement of chip scale atomic clocks to evaluate the viability
for such applications. Finally, we investigate the diffusion noise arising from the
xenon atoms adsorbed on the NEMS surface by putting a ~190 MHz
nanomechanical resonator into a phase-locked loop and measure the frequency
noise and Allan deviation.

4.1 Introduction
We have presented the theory of phase noise mechanism of NEMS in chapter 3.
So far, to our knowledge, none of fundamental noise sources proposed has been
measured and very little experimental results are available to decide whether the
predicted noise performance of NEMS can indeed be achieved. In this chapter, we
address this problem by inserting high Q NEMS resonators in phase-locked loops and
evaluate their noise performance against controlled variations in their environments.
We start our discussion from analyzing control servo behavior of a general phaselocked loop scheme based on NEMS and give the expressions for the locked condition
and loop bandwidth. We then present two electronic implementations of NEMS-based
phase-locked loops: (1) homodyne phase-locked loop based on a two port NEMS device
and (2) frequency modulation phase-locked loop (FM PLL). These phase-locked loops
are designed to lock minute electromechanical resonance of NEMS embedded in a large
electrical background as a result of diminishing transducer responsivity as the device
dimensions are scaled downward. The achieved noise floor in terms of phase noise
density and Allan deviation will be compared with the local oscillator (LO) requirement
of chip scale atomic clock (CSAC) to evaluate the viability of NEMS oscillators for this
application.1,2
Finally, we investigate the diffusion noise arising from the xenon atoms adsorbed
on the NEMS surface and measure the corresponding frequency noise and Allan
deviation using FM PLL. We will characterize the adsorption behavior, extract the
diffusion coefficients, and compare the experimental results with diffusion noise theory
and Yong and Vig’s model, both described in chapter 3.

4.2 Analysis of Phase-Locked Loop Based on NEMS
In general, two categories of schemes are commonly used for phase noise
measurement: self-oscillation and phase-locked loop (PLL). In the self-oscillation scheme
depicted in figure 4.1, the resonator operates within a positive feedback loop. The phase
noise, manifesting itself in the noise sideband around the carrier, is measured by a
spectrum analyzer (see section 2.3). The Allan deviation is calculated from the data taken
with the frequency counter. Such a scheme has widely been used to characterize
oscillators, and the detailed analysis can be found elsewhere.3
In this work, we extensively use the phase-locked loop scheme shown in figure
4.2(a). The principal elements of the loop are voltage control oscillator (VCO) and the
resonant response circuitry. The VCO is simply an oscillator whose frequency is
proportional to an externally applied voltage. The response function circuitry, containing
NEMS and phase detection circuitry, produces a quasi-dc signal proportional to the phase
of the mechanical resonance of NEMS. This phase sensitive signal is usually passed
through a loop filter, then applied to the control input of the VCO, and serves as error
signal to close the feedback loop. If the resonance frequency shifts slightly, the feedback
will adjust the control voltage to track the frequency change. Therefore, the voltage
fluctuation in the control input of the VCO reflects the frequency noise in the loop.
Moreover, the Allan deviation can be obtained from the data taken with the frequency
counter.

Spectrum
Analyzer
NEMS

Frequency
Counter

Figure 4.1. Self-oscillation scheme for the phase noise measurement of NEMS. The
principal components of the self-oscillation scheme are (1) NEMS, (2) the amplifier, and
(3) the phase shifter. The phase noise, manifesting itself in the noise sidebands around the
carrier, is measured by a spectrum analyzer. The Allan deviation can be calculated from
the data taken with the frequency counter.

(a)

KR
Spectrum
Analyzer

Frequency
Counter

VCO Control Voltage
(b)

NEMS

VCO
RF

Control Voltage

Figure 4.2. Configuration of a phase-locked loop based on NEMS. (a) Measurement
scheme of a phase-locked loop (PLL) based on NEMS. The principal components of a
PLL are the voltage controlled oscillator (VCO) and the resonant response circuitry
( K R ). The output of the resonant response circuitry is used as error signal to the control
input of the VCO to close the feedback loop. The frequency noise, manifesting itself as
voltage fluctuation in the control port of the VCO, is measured by a spectrum analyzer.
The Allan deviation is obtained from the data taken with frequency counter (C). (b)
Homodyne phase-locked loop. Homodyne phase-locked loop is one example of the
scheme shown in (a). In the homodyne phase detection, the NEMS device is driven by a
VCO at constant amplitude, and the output is amplified and mixed with the carrier. The
resonant response circuitry consists of NEMS, the amplifier, and the mixer.
71

We now analyze control servo behavior of the PLL, aiming to understand the
locked condition and the loop dynamics under the feedback. The frequency of the VCO is
determined by the control voltage Vcontrol , as given by

ωVCO = ωVCO (Vcontrol = 0) + K V Vcontrol .

(4.1)

K V and ωVCO (Vcontrol = 0) are the frequency pulling coefficient and the center frequency
of the VCO, respectively. The output of the resonant response circuitry can be
represented as a voltage function of carrier frequency, V R (ω C ) , and is applied to the
control input of VCO to provide the feedback. We analyze the loop behavior by
linearizing V R (ω C ) in the vicinity of the resonance frequency, ω 0 , of the NEMS as
V R (ω C ) = K R (ω 0 − ω C ) .

(4.2)

Here the proportional constant K R , called henceforth the resonant response coefficient,
is defined by K R = (∂
VR / ∂
ωC ) ω =ω . When the VCO is locked to the NEMS, we have the

condition VR = Vcontrol . From equation (4.1) and equation (4.2), we obtain the locked
condition

ω C = ωVCO (Vcontrol = 0) + K V K R (ω 0 − ω C ) .

(4.3)

We define the open loop gain of the PLL as
K loop = K V K R .

(4.4)

Therefore, equation (4.3) can be rewritten as

ωC =

K loop
1 + K loop

ω0 +

ωVCO (Vcontrol = 0) .
1 + K loop

(4.5)

Assuming that VCO is infinitely stable, i.e., ωVCO (Vcontrol = 0) is constant, equation
(4.5) implies that any frequency variation in the resonant frequency δω 0 of the device
72

will be scaled by a factor Kloop/(1+Kloop) as a result of feedback and reflected in the
corresponding carrier frequency change δωC in the phase-locked loop, i.e.,

δωC =

K Loop
1 + K Loop

δω 0 .

(4.6)

Equation (4.5) also implies an experimental way to measure the loop gain K loop .
We rewrite equation (4.5) as

ωC - ω0 =

(ωVCO (Vcontrol = 0) − ω 0 ) .
1 + K loop

(4.7)

In other words, ω C - ω 0 is proportional to ωVCO (Vcontrol = 0) − ω 0 with the proportionality
constant 1 /(1 + K loop ) . Experimentally one can hold the resonant frequency ω 0 constant,

rest the center frequency of VCO, ωVCO (Vcontrol = 0) , incrementally, and record the carrier
frequency ωC of the loop under lock. By plotting ω C versus ωVCO (Vcontrol = 0) , we can
determine the loop gain from the slope, i.e., the proportionality constant 1 /(1 + K loop ) .
So far we have considered the locked condition of the PLL in the steady state. We
now analyze the loop dynamics and give the expressions for the loop bandwidth. We first
discuss the case that a first-order low pass filter with a frequency cutoff, Δf filter , described
by the transfer function, H filter (ω ) = 1 /(1 + j (ω / 2πΔf filter )), is employed in the control input
of the VCO. Repeating the same steps from equation (4.1) to equation (4.5) by replacing
K V with K V H filter , we obtain

ωC =

K loop /(1 + K loop )

1 + jω /[(1 + K loop )2πΔf filter ]

ω0 +

1 + K loop H filter

ωVCO (Vcontrol = 0) .

(4.8)

Equation (4.8) means that the servo tracks the resonant frequency of the device with the
loop bandwidth Δf PLL given by
Δf PLL = Δf filter (1 + K loop ) .

(4.9)

Now we can write down the intrinsic bandwidth of the PLL limited by the NEMS
itself. This is done by simply replacing Δf filter in equation (4.9) with the resonant
bandwidth (ω 0 / 2πQ) in the loop.3 As a result, the intrinsic bandwidth of the PLL is given
by

Δf PLL = (ω 0 / 2πQ)(1 + K loop ) .

(4.10)

Both equation (4.9) and equation (4.10) imply that the effect of feedback enhances the
bandwidth by the factor 1 + K loop . For applications requiring fast response time, we can
always increase the loop gain to extend the loop bandwidth. Similar ideas have also been
used to enhance the bandwidth of atomic force microscopy by Albrecht et al.3
Finally, we give the explicit expression for the resonant response coefficient. The
resonant response function, V R (ω C ) , is determined by the transducer voltage from
NEMS, Vtransducer (ω C ) , cascaded by the amplifier gain, K A , and the gain of phase
detection circuitry, K P , as given by
V R (ω C ) = Vtransducer (ω C ) K A K P .

(4.11)

Taking the derivative of the resonant response function with respect to the carrier
frequency, the resonant response coefficient is given by
K R ≡(∂
VR / ∂
ω C ) ωC =ω0 = (∂Vtransducer / ∂ω ) ωC =ω0 K A K P .

(4.12)

As an example, we give the expression of resonant response coefficient for
homodyne phase-locked loop. The homodyne phase-locked loop shown in figure 4.2(b) is
74

one incarnation of the scheme shown in figure 4.2(a). In such a scheme, the NEMS
device is driven by a VCO at constant amplitude and the output is amplified and mixed
with the carrier. We first give the expression for the resonant response coefficient K R .
The gain of the phase detection circuitry K P is given by the mixer gain K M .
∂Vtransducer / ∂ω C can be approximated by

max
max
is the maximum
QVtransducer
/ ω 0 . Vtransducer

transducer voltage producing linear response. Thus, the loop gain of the PLL is given by
max
K R = K M K A QVtransducer
/ ω0 .

(4.13)

max
In the magnetomotive transduction, Vtransducer
is given by the electromotive force (emf)

voltage generated across the device with length L vibrating with the amplitude xC at the
frequency ω 0 in the magnetic field B , i.e.,
max
Vtransducer
= BLω 0 xC .

(4.14)

PLL

Frequency

Dimensions(L × w × t)

Meff

σA(1sec)

Δf

Two Port

125 MHz

1.6 μm × 800 nm × 70 nm 1300

1 pg

80 dB

4 × 10-7

165 kHz

FM VHF

133 MHz

2.3 μm × 150 nm × 70 nm 5000 100 fg

80 dB

5 × 10

32 Hz

FM VHF

190 MHz

2.3 μm × 150 nm × 100nm 5000 150 fg

80 dB

1 × 10-7

32 Hz

FM UHF

419 MHz 1.35 μm × 150 nm × 70 nm 1000

50 fg 100 dB

1 × 10

32 Hz

-8

-7

Table 4.1. Summary of parameters of all phase-locked loops based on NEMS
presented in this work

4.3 Homodyne Phase-Locked Loop Based upon a Two-Port NEMS Device

We now present the electronic implementation of the homodyne phase-locked
loop based on the scheme shown in figure 4.2(b) using a two port NEMS device, who
parameters are summarized in table 4.1. In practice, the two port topology avoids direct
electrical feedthrough of the simple one port scheme and allows careful design of the
bonding fixture to minimize the unwanted parasitic coupling that produces a large
electrical background on top of electromechanical resonance. Figure 4.3(a) shows the
SEM micrograph of the two port device fabricated from SiC epilayer with Au
metallization.4 It is driven magnetomotively and the resonant frequency is found to be
~125 MHz with quality factor Q =1300. Figure 4.3(b) shows the fundamental mode of
vibration of a two port device, optimized through the finite element simulation.
Figure 4.4 shows the electronic implementation of the PLL. The low phase noise
VCO (Minicircuits POSA-138) drives the NEMS device at constant amplitude and the
output of the transducer of the NEMS is amplified by a low noise preamplifier (Miteq
AU1442). We further employ an external bridge, consisting of a variable phase shifter
and a variable attenuator, to null out the electrical background. Figure 4.5 shows the
resulting mechanical resonant response of the NEMS after the nulling. The rising
background away from resonance shows the narrowband nature of the nullling, and
hence the locking range of the loop is limited within the natural width of the resonance
due the finite bandwidth of the variable phase shifter in the external bridge.The signal
from the external bridge is then mixed down with the carrier, amplified by an
instrumentation amplifier (Stanford Research Systems SR560), offset by a precision bias
circuit, and fed into the control input of the VCO to close the feedback control loop. Note
77

that the cutoff frequency of the low pass filter in the control servo is set to 1 MHz to fully
utilize the intrinsic bandwidth, (1 + K loop )(ω 0 / 2πQ) =165 kHz, provided by the NEMS
device (Kloop approximately equals to 1). Hence this scheme is very desirable in sensing
applications requiring fast response.
Figure 4.6 shows the phase noise spectrum of the VCO in PLL as measured by
spectrum analyzer (Hewlett Packard HP8563E). At frequencies between 100 Hz and 20
kHz, the spectrum exhibits flicker noise and has 1 / f 3 dependence on the offset
frequency due to the upconversion of the flicker noise of the preamplifier to the sideband
of the carrier. Above 20 kHz, the spectrum flattens out to about -110dBc/Hz, the
instrument noise floor of the spectrum analyzer.
Figure 4.7 shows the Allan deviation versus averaging times from frequency data
over the course of ~1000 sec interval taken with the frequency counter. At the
logarithmic scale, the observed Allan deviation, is nominally independent of averaging
time and confirms the flicker noise in the phase noise spectrum. Note that the error bar of
the each data point represents the confidence interval of the Allan deviation, given by

σ A (τ A ) / N S − 1 . For τ A = 1 sec, the observed Allan deviation σ A (τ A ) =4.7 x 10-7 is
consistent with the estimated value 7.7 x 10-7 from the theoretical expression

σ A (τ A ) = (1 / Q)10 - DR / 20 (with dynamic range DR =80 dB and Q=1300). In the present
experiment, DR is limited by (extrinsic) transducer-amplifier noise and the onset of the
Duffing instability of the NEMS device.

(a)

(b)

Figure 4.3. Pictures of two-port NEMS devices. (a) SEM micrograph of the two port
NEMS device. The device is fabricated from SiC epilayer with Au metallization. (b)
Finite element simulation of the fundamental mode of vibration of a two port device.

The two port device consists of two doubly clamped beams mechanically coupled by a
central beam. We use the finite element simulation to optimize the mechanical design.

Attn

NEMS

VCO
Offset
CTRL

LPF

RF
LO

Figure 4.4. Implementation of the homodyne phase-locked loop based on a two-port
NEMS device. We use a two port NEMS device with external bridge to implement the

homodyne phase-locked loop. The external bridge, comprised of a narrowband voltage
controlled phase shifter (φ) and a voltage controlled variable attenuator (Attn), is used to
null the electrical background.

350

Siganl Voltage (mV)

250

300

200
200

150
100

Phase (Degree)

400

300

100

50
120

122

124

126

128

130

Frequency(MHz)

Figure 4.5. Mechanical resonant response after nulling. The mechanical resonance of

the NEMS at 125 MHz is shown after the constant electrical background is nulled out by
an external bridge circuit. The rising background away from resonance shows the
narrowband nature of the nullling due to the bandwidth of the variable phase shifter in the
external bridge. This limits our locking range within the natural width of the resonance.

Phase Noise, Sφ (dBc/Hz)

-20
-40
-60
-80
-100
-120
10

Offset Frequency (Hz)

Figure 4.6. Phase noise density of the 125 MHz homodyne phase-locked loop based
on a two-port NEMS device. The phase noise density of the 125 MHz homodyne phase-

locked loop based on a two port NEMS is shown. Between 100 Hz and 20 kHz, the phase
noise spectrum exhibits flicker noise, i.e., having 1 / f 3 dependence on the offset
frequency. Above 20 kHz, it flattens out to ~110 dBc/Hz, the instrument noise floor of
the spectrum analyzer.

-6

Allan Deviation

-7

100

Averaging Time (sec)
Figure 4.7. Allan deviation of the 125 MHz homodyne phase-locked loop based on a
two-port NEMS device. The Allan deviation of 125 MHz homodyne phase-locked loop

versus averaging time, calculated from frequency data over the course of ~1000 sec
interval, is shown. At logarithmic scale, the Allan deviation is nominally independent of
averaging time and consistent with the observed flicker noise in the phase noise spectrum
in figure 4.6. The error bar in each data point represents the one-standard-deviation
confidence interval of the Allan deviation.

4.4 Frequency Modulation Phase-Locked Loop

We now present the analysis and implementation of the frequency modulation
phase-locked loop, which is designed to lock the even smaller electromechanical
resonance of a NEMS embedded in a large electrical background. Roughly speaking, the
frequency modulation of the carrier and subsequent demodulation by lock-in detection
after mixer generates an electrical signal proportional to the derivative of the resonant
response with respect to frequency. As a result, the constant electrical background, in
which the electromechanical resonance of the NEMS is embedded, is nulled out. As
shown in figure 4.8, the FM PLL is formed by adding frequency modulation of the carrier
and lock-in detection to the homodyne phase-locked loop. One can prove that addition of
the frequency modulation and lock-in detection contributes to K R with two additional
gain factors, the frequency modulation index Μ and the lock-in detection gain K Lockin .
By inserting these two factors into equation (4.13), we find
max
K R = ( K M K A QVtransducer
/ ω 0 )Μ K Lockin .

(4.15)

Note that Μ = Vm K V / ω m in the case that a sinusoidal voltage of magnitude Vm at
modulation frequency, ω m is applied to the control voltage port of VCO.

NEMS

Lock-in

VCO
RF

Control Voltage

Figure 4.8. Conceptual diagram of frequency modulation phase-locked loop (FM
PLL) scheme. Similar to homedyne phase-locked loop, the NEMS is driven by a VCO

at constant amplitude, and the output is amplified and mixed with the carrier. The FM
PLL is formed by adding the frequency modulation (FM) of the carrier and the lock-in
detection to the homodyne phase-locked loop.

Figure 4.9 shows the electronic implementation of the FM PLL at VHF (very high
frequency) band for 133 MHz and 190 MHz devices. The device configurations are
summarized in table 4.1. We detect the mechanical resonance in the reflection scheme by
a directional coupler. The signal from the NEMS device is amplified by a radio frequency
(RF) amplifier with gain, K RF , shifted in phase by the phase shifter, and subsequently
mixed down to intermediate frequency (IF) by a mixer. The IF signal is further amplified
by an IF amplifier with gain, K IF . The total amplifier gain is given by
K A = K RF K IF .

(4.16)

In our experiment, the carrier is modulated at 1.2652 kHz with reference oscillator
in the lock-in amplifier. The lock-in amplifier (Stanford Research Systems SR830) is
employed to detect the signal amplitude at the modulation frequency and subsequently
rescale the readout according to the Sensitivity setting with full scale voltage V fullscale .
This is further divided by a voltage divider with a dividing factor DF. For convenience,
we incorporate the voltage division and rescaling into the lock-in detection gain

K lockin = (1 / DF )V fullscale / Sensitivity .

(4.17)

We summarize the experimental parameters used in FM PLL at 190 MHz in table 4.2. To
close the feedback loop, the lock-in amplifier outputs the signal to the control port of the
VCO. We use the frequency synthesizer (Hewlett Packard HP8648) in frequency
modulation mode as the VCO. This imposes a proportional control with a frequency
cutoff proportional to the inverse of the lock-in time constant τ lockin . More precisely, the
bandwidth of the FM PLL is given by Δf PLL = (1 + K loop )(1 / 2πτ lockin ) using equation (4.9)
as a result of feedback. In addition, a digital feedback loop is established by a computer
86

interface, which periodically checks the digital output of the lock-in amplifier, computes
a correction, and resets the center frequency of the VCO accordingly with prescribed loop
gain and loop time. Effectively one has a discrete integral control, extending the locking
range beyond the natural width of the resonance. We use it to follow the frequency shift
induced by large changes in device mass over extended measurement intervals.

Figure 4.9. Implementation of frequency modulation phase-locked loop (FM PLL)
scheme. We employ frequency modulation phase-locked loop (FM PLL) scheme to track

the resonant frequency of the device. The mechanical resonance is detected in a reflection
scheme, by using a directional coupler (CPL). The reflected signal is amplified, phase
shifted (Φ), and mixed down (⊗). We modulate the carrier at 1.2652 kHz and employ a
lock-in amplifier (Stanford Research Systems SR830) for demodulation. The resulting
output (X) provides the analog feedback to the VCO (Hewlett Packard HP8648B). A
computer-controlled parallel digital feedback (µC) is implemented for applications
requiring a large locking range.

Figure 4.10 shows the phase noise spectrum of the FM PLL based on the 190
MHz device. At frequencies between 15 mHz and 30 Hz, the spectrum exhibits flicker
noise, having 1 / f 3 dependence on the offset frequency. Above 30 Hz, the spectrum rolls
off at the slope of 50 dB/decade, reflecting the loop bandwidth limited by the filter in
lock-in detection. With the lock-in time constant τ lockin = 10 ms and K loop =1, we estimate
the loop bandwidth to be 32 Hz from the expression Δf PLL = (1 + K loop )(1 / 2πτ lockin ) .
Figure 4.11 shows the observed Allan deviation calculated from data over the course of
one hour taken with frequency counter (Agilent 53132) for τ A ranging from 1 sec to 128
sec. At the logarithmic scale, the observed Allan deviation versus averaging time is
nominally constant and thus consistent with flicker noise in the phase noise spectrum in
Figure 4.10. The Allan deviation σ A (τ A ) =1x10-7 for τ A = 1 sec is close to the estimated
value 2x10-7 from the expression σ A (τ A ) = (1 / Q)10 - DR / 20 (with dynamic range DR = 80
dB and Q= 5000).
We have also implemented the FM PLL at the ultra high frequency (UHF) band at
419 MHz. For the 419 MHz SiC device with dimensions 1.35 um(L) x 150 nm(w) x 70
nm (t) and Q=1600, the mechanical impedance is only ~0.08 Ω and embedded in a large
electrical impedance ~100 Ω. To detect such a small impedance at the UHF band, we
replace the simple reflection scheme used for FM PLL at the VHF band with a balanced
bridge detection (see also chapter 7).5 To amplify the signal from the NEMS device, we
employ a cryogenic amplifier (Miteq AFS3-00100200-09-CR-4), working from 0.1 to 2
GHz with the equivalent noise temperature TN =10 K at 419 MHz. We summarize the
experimental parameters in table 4.2. Figure 4.12 shows the observed phase noise
89

spectrum. At frequencies between 15 mHz and 30 Hz, the spectrum exhibits 1 / f 2
dependence on the offset frequency. Similar to the FM PLL at VHF band, the additional
rolloff

the

spectrum

results

from

the

loop

bandwidth

Δf PLL = (1 + K loop )(1 / 2πτ lockin ) =32 Hz (with τ lockin = 10 ms and K loop =1). Figure 4.13
shows the Allan deviation calculated from the frequency data over the course of one hour
taken with the frequency counter (Agilent 53132). For τ A =1 sec, the observed Allan
deviation σ A =1 x 10-7 is much higher than the estimated value of 6.25 x 10-9 from the
expression σ A (τ A ) = (1 / Q)10- DR / 20 with DR=100 dB and Q=1000. (DR is estimated
assuming the white noise contribution from the cryogenic amplifier with equivalent noise
temperature TN =10 K at 419 MHz and onset of Duffing nonlinearity of the NEMS.) We
attribute this discrepancy to the conversion of other noise sources to side band of the
carrier through the mixer or the nonlinearity of the circuit.

Table 4.2. Summary of experimental parameters used in the frequency modulation
phase-locked loops (FM PLL) at very high frequency (VHF) and ultra high
frequency (UHF) bands

Parameter

Symbol

VHF PLL

UHF PLL

Resonant Frequency

ω0/2π

190MHz

419MHz

Quality Factor

5000

1600

Transducer Voltage

max
Vtransucer

~1μV

RF Gain

KRF

35dB

45dB

IF Gain

KIF

500

Mixer Gain

~0.1

Modulation Frequency

ωm/2π

1.3kHz

10kHz

Modulation Voltage

30mV

Frequency Pulling
Coefficient

100kHz/V

50kHz/V

Sensitivity

200mV

1 mV

Dividing Factor

Full Scale Voltage

Vfullscale

10V

Lock-in Time Constant

τlock-in

10ms

Modulation Index

2.3

0.15

Loop Gain (Estimated)

Kloop

~4.8

~4.5

Loop Gain (Measured)

Kloop

Phase Noise, Sφ (dBc/Hz)

100

-50

-2

-1

Offset Frequency (Hz)
Figure 4.10. Phase noise density of the 190 MHz frequency modulation phase-locked
loop (FM PLL). The phase noise density of the 190 MHz FM PLL is shown. Between 15

mHz and 30 Hz , the phase noise spectrum of the 190 MHz phase-locked loop exhibits
flicker noise, i.e., having 1 / f 3 dependence on the offset frequency. Above 30 Hz, it rolls
off at the slope of 50 dB/decade due to the loop bandwidth limited by lock-in detection

Δf PLL = (1 + K loop )(1 / 2πτ lock − in ) =32 Hz (with τ lock − in =10 ms and K loop =1).

Allan Deviation

-7

-8

100

Averaging Time (sec)
Figure 4.11. Allan deviation of the 133 MHz frequency modulation phase-locked
loop (FM PLL). The Allan deviation of the 133 MHz FM PLL versus averaging time,

calculated from frequency data over the course of one hour, is shown here. At logarithmic
scale, the Allan deviation is nominally independent of averaging time and thus consistent
with the observed flicker noise in the phase noise spectrum. The error bar in each data
point represents the one-standard-deviation confidence interval of the Allan deviation.

Phase Noise, Sφ(dBc/Hz)

100
80
60
40
20
-20
-40
-2

-1

Offset Frequency(Hz)
Figure 4.12. Phase noise density of the 419 MHz frequency modulation phase-locked
loop (FM PLL). The phase noise density of the 419 MHz FM PLL is shown. Between 15

mHz and 30 Hz, the phase noise spectrum of the 419 MHz phase-locked loop exhibits
white noise, having 1 / f 2 dependence on offset frequency. Above 30 Hz, it rolls off at the
slope of

40 dB/decade due to the loop bandwidth limited by lock-in detection

Δf PLL = (1 + K loop )(1 / 2πτ lock − in ) =32 Hz (with τ lock − in =10 ms and K loop =1).

-6

Allan Deviation

-7

100

Averaging Time (sec)
Figure 4.13. Allan deviation of the 419 MHz frequency modulation phase-locked
loop (FM PLL). The Allan deviation of the 419 MHz FM PLL versus averaging time,

calculated from frequency data over the course of one hour, is shown here. The error bar
in each data point represents the one-standard-deviation confidence interval of the Allan
deviation.

4.5 Comparison with Local Oscillator Requirement of Chip Scale Atomic
Clock

The chip scale atomic clock is the vapor cell atomic clock, scaled down to
microelectronic chip size.6 The operation of CSAC requires a LO to interrogate the
atomic transitions to provide the frequency precision. The frequency reference
configuration consists of the physics package, the control circuitry, and the LO. In the
physics package, the hyperfine transition of the atoms in the vapor cell is induced by a
vertical cavity surface emitting laser (VCSEL), modulated at microwave frequency. The
optical transmission is subsequently sensed by a semiconductor detector to produce a
microwave signal, which is phase locked to the LO to optimize the long term and short
term frequency stabilities through the control circuitry. Due to their small size and low
operating power, NEMS oscillators are very promising candidates as the LO for CSAC,
so it is interesting to compare our achieved noise floor with the LO requirement and
evaluate their viability for such applications.
Kitching calculates the LO requirement by demanding that the fractional
frequency instability of the CSAC satisfy the DAPRA program goal of 10-11 for one hour
averaging time.5 Figure 4.14 shows the phase noise floor of the LO requirement using the
hyperfine transitions of Rb87 at frequency 6.8 GHz together with the measured phase
noise spectra of all the phase-locked loops presented so far, properly scaled to 6.8 GHz.
Although the high frequency (>20 kHz) and low frequency (<0.5 Hz) ends of the spectra
barely meet the requirement, the middle band between 0.5 Hz and 20 kHz is still 40 dB
higher than the requirement. This is due to extrinsic transducer amplifier noise in our still
unoptimized experimental configuration. Also shown in figure 4.14 are the projected
phase noise spectra of 400 MHz NEMS-based oscillators with Q = 104 and Q = 105,
96

limited by thermomechanical noise at room temperature. They are certainly able to meet
the requirements of CSAC with at least 30 dB margin at all frequencies.
Figure 4.15 shows the corresponding Allan deviations of all phase-locked loops
versus averaging time τ A , and the LO requirement. For τ A longer than 1 sec, all the
experimentally achieved Allan deviations already meet the LO requirement. For τ A < 1
sec, the only available Allan deviation data for the 419 MHz PLL exhibits 1 / τ A
dependence on the averaging time, which is worse than the LO requirement. Also shown
in figure 4.15 are the projected Allan deviations of 400 MHz NEMS-based oscillators for

Q=104 and Q=105, limited by thermomechanical noise at room temperature. They are
certainly able to meet the LO requirement for all averaging times ranging from 10-7 sec to
100 sec. Meeting the LO requirement in terms of both phase noise spectra and Allan
deviations clearly demonstrate the viability of the NEMS oscillators as the LO for CSAC.

CSAC Local Oscillator
Stability Requirements
87
( 6.8 GHz Rb Clock )

Phase Noise, S φ (dBc/Hz)

150

420MHz NEMS, Q~1200
Scaled to 6.8GHz Low Frequency Phase Noise
from a 190MHz NEMS, Q=5000
Scaled to 6.8 GHz
Phase Noise Data for a
124MHz NEMS, Q=1300
Scaled to 6.8 GHZ

100

Q=10

50
-50
-100
-150

Q=10

Projections for "nextgen" 400MHz NEMS
scaled to 6.8 GHZ
Thermomechanical Limit
Practical, Readout Limited
Thermomechanical Limit
Practical, Readout Limited
-3

-2

10 10 10

-1

10 10 10 10 10 10 10 10

Offset Frequency (Hz)
Figure 4.14. Phase noise spectrum of NEMS-based phase-locked loops versus the
local oscillator (LO) requirement of chip scale atomic clock (CSAC). The measured

phase noise spectra of 125 MHz, 190 MHz and 419 MHz phase-locked loops based on
NEMS are compared to the LO requirement of CSAC, upon proper scaling to 6.8 GHz.
The projected phase noise spectra of 400 MHz NEMS oscillators with Q=104 and Q=105,
limited by thermomechanical fluctuations at room temperature, clearly shows the ability
to meet the CSAC requirement.

-5

CSAC LO
Stability Requirements
-6

124MHz NEMS
Q=1300

Allan Deviation

420MHz NEMS
Q~1200

-7

133MHz NEMS
Q=5000

-8

Solid: Thermomechanical Limit
Dash: Practical Limit in Experiments

-9

Q=10

400MHz NEMS
Projection

Q=10

-10

-7

-6

-5

-4

-3

-2

-1

10 10 10 10 10 10 10

10 10 10 10 10

Averaging Time (sec)
Figure 4.15. Allan deviations of NEMS-based phase-locked loops versus the local
oscillator (LO) requirement of chip scale atomic clock (CSAC). The measured Allan

deviations of 125 MHz, 190 MHz and 419 MHz phase-locked loops based on NEMS are
compared to the LO requirement of CSAC. The projected Allan deviations of 400 MHz
oscillators based on NEMS of Q=104 and Q=105, limited by thermomechanical
fluctuations at room temperature, clearly meet the LO requirement of CSAC.

4.6 Experimental Measurement of Diffusion Noise

We have analyzed many noise processes in detail in chapter 3. All these noise
processes arise from local fluctuation of the intrinsic thermodynamic properties of a
physical system in equilibrium.7 Although these fluctuations become noise which limits
NEMS performance as sensors or resonators, they also provide a potential source of
information.8,9 The fluctuation around the thermodynamic mean is proportional to the
number of independent accessible degree of freedom. Moreover, the spectral density of
fluctuations is precisely governed by the dynamic parameters of the systems as generally
expressed by the fluctuation-dissipation theorem.7 In particular, when gaseous species
adsorb on a NEMS device, typically from the surrounding environment, they can diffuse
along the surface in and out of the device. Thus the number of adsorbed atoms in the
device can fluctuate, which translates into mass fluctuation and hence frequency
fluctuations. The noise spectrum in this case is governed by the diffusion time. We have
proposed the diffusion noise theory of NEMS in section 3.5. Here we demonstrate the
experimental measurement of the diffusion noise arising from adsorbed xenon atoms on
NEMS surface.
We incorporate the NEMS device into a low-noise FM PLL circuit (see section
4.4). Data are obtained from a NEMS resonator with fundamental resonant frequency

f0~190 MHz and dimensions, 2.3 μm (L) × 150 nm (w) × 100 nm (t). (The surface area of
the device is AD =3.45x10-13 m2.) The device is a doubly clamped beam patterned from
SiC epilayers4 and capped with thermally evaporated dual metallic layers: 30 nm Al
(bottom) and 5 nm Ti (top). (The effective vibratory mass of the device, including the
metallic layers, is M eff =96 fg.) After fabrication, the device is loaded into a UHV
100

cryostat, actuated magnetomotively,4 and exhibits Q of ~5000 for the fundamental inplane flexural mode of vibration at the measurement temperatures ~58 K.
Xenon is used in our experiments due to its large atomic mass ( m Xe =130 amu),
and its well characterized surface behavior in literature.10-14 A gas nozzle is used to
deliver a constant, calibrated flux to the device (see also section 5.2). The flux to the
device, Φ , is 2.65×1017 atom/m2sec, corresponding to an effective pressure of 6.6×10-8
torr at 58 K. Data presented here are taken at constant flux, while changing the
temperature of the device.

101

Nozzle

NEMS

Figure 4.16. Experimental configuration for diffusion noise measurement. A gas

nozzle with a 300 μm aperture provides a controlled flux of atoms or molecules. The
flux is determined by direct measurements of the gas flow rate, in conjunction with a
well-validated model for the molecular beam emanating from the nozzle.

102

First, we measure the adsorption spectrum, defined as the total number of
adsorbed xenon atoms versus temperature. As the device is cooled below 57 K, we
observe irreversible accumulation of xenon in solid phase due to the two-dimensional
solid-gas phase transition.11 The adsorption of xenon is fully reversible above this
transition temperature. All the measurements are thus done above 57 K. We take the
resonance frequency data of the device versus temperature with applied flux and without
flux, denoted by f G (T ) and f NG (T ) , respectively. The adsorption spectrum is deduced
from the frequency change by N (T ) = − m Xe ( f G (T ) − f NG (T ) /(ℜ / 2π ) due to the
presence of gas, where ℜ / 2π = f 0 / 2M eff = 0.99 Hz/zg is the mass responsivity of the
device.15 The coverage θ , defined as the number of adsorbed atoms per unit area, i.e.,
N (T ) / AD , is 6.67×1014 atoms/cm2 at T =58 K, consistent with a commensurate

monolayer coverage of

5×1014 atoms/cm2 on Pt(111) at T=85 K.13 Because the

roughness of thermally evaporated Ti top layer of the device completely blurs the
monolayer transition of xenon, we do not observe such a transition in the adsorption
spectrum.2

103

2.0
1.5

N(T) (x10 atoms)

2.5

1.0
0.5
0.0

T (K)
Figure 4.17. Adsorption spectrum of xenon atoms on NEMS surface. The adsorption

spectrum is deduced from N (T ) = − m Xe ( f G (T ) − f NG (T ) /(ℜ / 2π ) , ℜ / 2π = 0.99 Hz/zg

is the mass responsivity of the device. f G (T ) and f NG (T ) denote the resonant frequency
data with applied gaseous flux and no flux, respectively.

104

Figure 4.18 shows the representative data of the spectral density of fractional
frequency noise at T= 58 K with and without gaseous flux. The spectrum with no applied
flux exhibits flicker noise from 0.1 Hz to 2 Hz and flattens out above 2 Hz, reflecting the
instrumentation noise of FM PLL. In contrast, the spectrum with applied flux clearly
shows excess noise contribution from gas. We have of course tested that the parameters
affecting the loop gain of FM PLL (in particular, the quality factor of the resonator) do
not change with temperature or coverage, and therefore cannot be responsible for the
excess noise. More quantitatively, we calculate the fractional frequency noise contributed
from gas, S yG (ω ) , by subtracting the spectral density of fractional frequency noise with
zero flux, S yNG (ω ) , from that with applied flux at given temperatures S Total
(ω ) , i.e., from
(ω ) − S yNG (ω ). All the resulting spectra, as shown in figure
the formula S yG (ω ) = S Total

4.19, exhibit predicted functional form from equation (3.53), i.e.,
S y (ω ) =

aN (T ) m Xe 2
cos ωτ
) ∫
dτ .
4π
M eff 0 (1 + τ / τ D )1 / 2

(4.18)

These spectral data are fitted to extract the diffusion time τ D , using equation (4.18).
Because the extracted diffusion times, ranging from 0.114 sec to 0.053 sec, are much
shorter than the typical correlation times of an adsorption-desorption cycle,16 the
observed noise spectra cannot be attributed to adsorption-desorption process.

105

-6

-7

1/2

10
Gas
No Gas

1/2

SM (ω) (zg/ Hz )

1/2

Sy (ω) (1/ Hz )

100

-8

ω/2π (Hz)
Figure 4.18. Representative fractional frequency noise spectra. The spectral density

of fractional frequency with and without gaseous flux at T=59.2 K is shown. The
spectrum, S yNG (ω ), with no applied flux (black) reflects the instrumentation noise of FM
(ω ) with applied flux (red) clearly shows excess
PLL. In contrast, the spectrum S Total

noise contribution from gas. The right hand axis shows the scale of the corresponding
mass fluctuations in unit of zg/Hz1/2.

106

-6

1/2

SM (ω) (zg/ Hz )

1/2

Sy (ω) (1/ Hz )

100

-7

-8

58K
59K
60.7K
63.4K

1/2

ω/2π (Hz)
Figure 4.19. Spectral density of fractional frequency noise contributed from gas. The

spectral density of fractional frequency noise contributed from the gaseous flux at four
measurement temperatures is displayed. We calculate the fractional frequency noise
contributed from gas, S yG (ω ) , by subtracting the spectral density of fractional frequency
noise with zero flux, S yNG (ω ) , from that with applied flux at given temperatures
S Total
(ω ) , i.e., from the formula S yG (ω ) = S Total
(ω ) − S yNG (ω ) . The right hand axis shows

the scale of the corresponding mass fluctuations in unit of zg/Hz1/2.

107

From the diffusion noise theory, we can also calculate the diffusion coefficients

D by D = L2 /(2a 2τ D ) , where a =4.43 is a numerical factor, and L =2.3 μm is the device
length, (see section 3.5). Table 4.3 lists the extracted diffusion times and diffusion
coefficients together with the corresponding coverage at four measurement temperatures.
In general, the surface diffusion of xenon is determined by the corrugation of the
adsorbate-surface potential and the attractive interactions between the adsorbed atoms.13
At very dilute limits, the xenon atoms behave and diffuse like an ideal two-dimensional
gas.10,12 At higher coverage, however, the diffusion is more dominated by the attractive
interaction between the adsorbed xenon atoms and as a result, the diffusion coefficient
dramatically decreases with increasing coverage.13 Our extracted diffusion coefficients
are very close to D=2x10-8 cm2/s, reported by Meixner and George for xenon on Pt(111)
for coverage θ = 5x1014 atoms/cm2 in spite of very different surface conditions and
measurement temperatures.13 The indifference of the diffusion coefficients to surface
conditions and temperatures suggests that in both cases the attractive interaction between
adsorbed xenon atoms completely dominates the surface diffusion.
Figure 4.21 shows the measured Allan deviation σ A (τ A ) versus temperature with
and without the gaseous flux for averaging time τ A = 1 sec. The Allan deviation with zero
applied flux, denoted by σ ANG , reflects the instrumentation noise floor of the FM PLL and
slightly decreases with temperature. We attribute this slight decreasing trend with
temperature to the corresponding increase in quality factor (15%) from T=75 K to T=58
K. In contrast, for temperature below 65 K, the Allan deviation with gaseous flux, σ Total
clearly exceeds the instrumentation noise floor due to the excess noise contribution from
the gas. Figure 4.22 shows the Allan deviation contributed from gas, σ AG , calculated by
108

subtracting the Allan deviation without gas from Allan deviation with gas, i.e., from the
) 2 − (σ ANG ) 2 .
formula (σ AG ) 2 = (σ Total
From equation (3.58), the expression for Allan deviation from diffusion noise
theory is

2aN (T ) ⎛⎜ m Xe ⎞⎟
sin
σ (τ A ) = ∫
ωτ
Χ ( D ),
τA
π ⎝ M eff ⎠
0 (ωτ A )

(4.19)

where Χ (τ D / τ A ) is a function defined in equation (3.59). Equation (4.19) shows that
Allan deviation associated with the number fluctuation of an ensemble of adsorbed atoms
is proportional to the square root of its total number, N (T ) . Roughly speaking, we can
thus attribute the monotonically increasing trend in Allan deviation in figure 4.22, as
temperature is lowered, to the corresponding increase in the number of adsorbed xenon
atoms in figure 4.17. Using equation (4.19) and measured N (T ) and τ D from table 4.3,
we calculate the Allan deviation and display the result in figure 4.22. In figure 4.22, we
also show the calculated Allan deviation, σ A (τ A ) = N a σ OCC (mXe / M eff ) τ A / 6τ r , from
Yong and Vig’s model for the case of immobile monolayer adsorption, assuming the
monolayer coverage N a = 2.3x106 at T=58 K and the sticking coefficient s=1 to estimate
the correlation time for an adsorption-desorption cycle from τ r = N (T ) /(ΦsAD ) and the
= N ( N a − N ) / N a2 (see section 3.4).17,18
variance of occupational probability from σ OCC

As shown in figure 4.22, the experimentally observed Allan deviation is consistent with
the prediction from diffusion noise theory. In contrast, the estimated Allan deviation from
Yong and Vig’s model, vanishing at completion of one monolayer, is apparently
contradictory to experimental observation.
109

(a)

(b)

-12

-13

Sy(ω) (1/Hz)

-14

-15

-16

-17

-15

-16

-17

ω/2π (Hz)

(c)

(d)

-12

-13

Sy(ω) (1/Hz)

-14

-15

-16

-17

ω/2π (Hz)

Sy(ω) (1/Hz)

-14

-15

-16

-17

ω/2π (Hz)

Figure 4.20. Spectral density of fractional frequency noise with fitting. Data (black)

from figure 4.19 are fitted by a predicted function form in equation (4.18) (red) from
diffusion noise theory to extract the diffusion times. (a) Spectral density data with
fitting at T= 58 K. (b) Spectral density data with fitting at T=59.2 K. (c) Spectral
density data with fitting at T=60.7 K. (d) Spectral density data with fitting at T=63.4
K.

110

Temp

τD

atom

atom/cm

sec

cm2/sec

2.30× 106

6.67× 1014

0.114

1.15 × 10-8

59.2

1.79× 106

5.19× 1014

0.0637

2.06 × 10-8

60.7

1.33 × 106

3.86 × 1014

0.0553

2.37 × 10-8

63.4

8.08× 105

2.34× 1014

0.0530

2.47 × 10-8

Table 4.3. Summary of diffusion times and coefficients versus temperature

111

200

10
With Gas
No Gas

150

100

δM (zg)

-7

σA (x10 )

T (K)
Figure 4.21. Allan deviation data with gas and without gas. The Allan deviation

(black) with zero applied flux reflects the instrumentation noise floor of the FM PLL. For
temperature below 65 K, the Allan deviation (red) with gas clearly exceeds that without
gas due to the excess noise contribution from the gas. The right-hand axis shows the scale
of the corresponding mass fluctuation in units of zg.

112

12
200
Experiment
Diffusion
Yong and Vig

150

-7

σA(x10 )

100

δM(zg)

T (K)
Figure 4.22. Comparison with prediction from diffusion noise theory and Yong and
Vig’s model. The Allan deviation (red) contributed from gas, σ AG , is calculated by

subtracting the Allan deviation without gas from Allan deviation with gas, i.e., from the
) 2 − (σ ANG ) 2 . The Allan deviation (blue) from diffusion noise is
formula (σ AG ) 2 = (σ Total
calculated using equation (4.19) and measured N (T ) and τ D from table 4.3. For
comparison, the calculated Allan deviation (dark gray) from Yong and Vig’s model is
also displayed, assuming the monolayer coverage N a = 2.3x106 at T=58 K and the
sticking coefficient s=1. The right hand axis shows the scale of the corresponding mass
fluctuation in units of zg.

113

As already mentioned, no appreciable change in quality factor is observed from
the adsorbed species in our experiment and thus the observed diffusion noise is nondissipassive in nature, a very important attribute of parametric noise as pointed out by
Cleland and Roukes.17
Having verified that the observed fluctuations are due to the mass fluctuation
caused by diffusion, we can relate the spectral density of mass fluctuation S M1 / 2 (ω ) to the
spectral

density

fractional

frequency

noise

S 1y / 2 (ω )

the

expression

S M1 / 2 (ω ) = f 0 S 1y / 2 (ω ) /(ℜ / 2π ) with the mass responsivity. Similarly, we relate the Allan

deviation to the corresponding mass fluctuation δM by δM = f 0σ A /(ℜ / 2π ) . The scale
in the right hand axes in figure 4.18, 4.19, 4.21 and 4.22 shows that the corresponding
mass fluctuation is on the order of tens of zeptogram (1 zeptogram = 10-21 gram) and thus
our experiment is indeed the “fluctuation sensing” at zeptogram scale.

4.7 Conclusion

In this chapter, we present the experimental measurement of phase noise of
NEMS. We first analyze control servo behavior of a phase-locked loop based on NEMS
and give the expressions for the locked condition and measurement bandwidth. Based on
such a scheme, we then present in detail several electronic implementations, all of which
are designed to lock minute mechanical resonance of NEMS and complement each other
in their merits. The homodyne phase-locked loop based on a two-port NEMS device fully
utilizes the intrinsic bandwidth provided by NEMS, and is very desirable for sensing
applications requiring fast response time. It requires, however, laborious manual
adjustments and is limited in the locking range. On the other hand, the FM PLL, touted
114

for its easy loop implementation and large locking range, suffers from the limited
bandwidth due to the lock-in detection. In general, the observed Allan deviation σ A (τ A )
is consistent with the estimated value from the expression σ A (τ A ) = (1 / Q)10- DR / 20 with
experimentally determined dynamic range DR and Q. We summarize the performance of
all the phase-locked loops with their device parameters considered in this chapter in table
4.1.
We then consolidate the phase noise and Allan deviation data of all the phaselocked loops and compare them with the LO requirement of CSAC. While our current
performance, limited by transducer amplifier noise, only partially meets the requirement,
the projected phase noise and Allan deviation for 400 MHz NEMS based oscillators with

Q=104 and Q=105, limited by thermomechanical noise, clearly show the potential for this
application.
We further demonstrate the measurement of diffusion noise arising from adsorbed
xenon atoms on the NEMS device. In general, our experimental results can be explained
with the diffusion noise theory. The measured spectra of fractional frequency noise
confirm the predicted functional form from the diffusion noise theory and the extracted
diffusion coefficients agree well with the reported values in literature. Moreover, the
measured Allan deviation contributed from gas is consistent with the theoretical estimates
from diffusion noise theory, using the total number of adsorbed atoms and extracted
diffusion times. Finally, we point out that the diffusion noise or its equivalent mass
fluctuation, measured with unprecedented mass sensitivity at zeptogram level, imposes
the ultimate sensitivity limits of any nanoscale gas sensors, regardless of their sensing
mechanisms. But more importantly, this work, for the first time, goes beyond simple
115

measurement of adsorption spectrum in nanodevices and demonstrate a canonical
example where a high quality factor NEMS device, inserted into a phase-locked loop,
serves to probe the noise process of the adsorbed species and extract the microscopic and
dynamic information of surface diffusion. We expect the generalization of this approach
will find many interesting applications in surface science of nanodevices.

116

References

H. Pauly and G. Scoles (editor) Atomic and Molecular Beam Methods (New
York, Oxford University Press, 1988).

J. Krim, D. H. Solina, and R. Chiarello Nanotribology of a Kr Monolayer: a
quartz-crystal microbalance study of atomic scale friction. Phys. Rev. Lett. 66,
181-184 (1991).

T. R. Albrecht, P. Grutter, D. Horne, and D. Rugar Frequency modulation
detection using high Q cantilever for enhanced force microscopy sensitivity. J.
Appl. Phys. 69, 668 (1991).

Y. T. Yang, K. L. Ekinci, X. M. H. Huang, M. L. Roukes, C. A. Zorman, and M.
Mehregany Monocrystalline 3C-SiC nanoelectromechanical systems. Appl. Phys.
Lett. 78, 612 (2001).

J. Kitching Local oscillator requirements for chip-scale atomic clocks. Private
communication (2004).

J. Kitching, S. Knappe, and L. Hollberg Miniatured vapor-cell atomic frequency.
Appl. Phys. Lett. 81, 553 (2002).

L. D. Landau and E. M. Lifshitz Statistical Physics Vol. 1 (Oxford, ButterworkthHeinemann,1980).

E. L. Elson and D. Magde Fluoresence correlation spectroscopy I concept basis
and theory. Biopolymer 13, 1-27 (1974).

D. Magde, E. L. Elson, and W. W. Webb Thermodynamic fluctuations in a
reacting system- Measurement by fluorescence correlation spectroscopy. Phys.
Rev. Lett. 29, 705-708 (1972).

10.

C. T. Rettner, D. S. Bethune, and E. K. Schweizer Measurement of Xe desorption
rates from Pt(111):Rates for an ideal surfaces and in the defect-dominated regime.
J. Chem. Phys. 92, 1442 (1990).

11.

H. Clark The theory of Adsorption and Catalysis (London, Academic Press,
1970).

12.

J. Ellis, A. P. Graham, and J. P. Toennies Quasielastic helium atom scattering
from a two-dimensional gas of xenon atoms on Pt(111). Phys. Rev. Lett. 82, 50725075 (1999).

13.

D. L. Meixner and S. M. George Surface diffusion of xenon on Pt(111). J. Chem.
Phy. 98, 11 (1983).
117

14.

H. J. Kreuzer and Z. W. Gortel Physisorption Kinetics (Heidelberg, SpringerVerlag, 1986)

15.

K. L. Ekinci, Y. T. Yang, and M. L. Roukes Ultimate limits to inertial mass
sensing based upon nanoelectromechanical systems. J. Appl. Phys. 95, 2682
(2004).

16.

To measure the correlation time of an adsorption desorption cycle, we dose the
device with given coverage, block the gaseous flux with mechanical shutter, and
observe the coverage over the time. Under similar conditions, the typical
correlation time, obtained by fitting the exponential decay of the coverage, is >2
sec.

17.

A. N. Cleland and M. L. Roukes Noise processes in nanomechanical resonators.
J. Appl. Phys. 92, 2758 (2002).

18.

Y. K.Yong and J. Vig Resonator surface contamination: a cause of frequency
fluctuations. IEEE Trans. On Ultrasonics, Ferroelectric, and Frequency Control
36, 452 (1989).

118

Chapter 5
Zeptogram Scale Nanomechanical
Mass Sensing

Very

high

frequency

nanoelectromechanical

systems

unprecedented sensitivity for inertial mass sensing.

(NEMS)

provide

We demonstrate in situ

measurements in real time with mass noise floor ~20 zeptogram. Our best mass
sensitivity corresponds to ~7 zeptograms, equivalent to ~30 Xenon atoms or the
mass of an individual 4 kDa molecule. Detailed analysis of the ultimate sensitivity of
such devices based on these experimental results indicates that NEMS can
ultimately provide inertial mass sensing of individual intact, electrically neutral
macromolecules with single-Dalton (1 amu) sensitivity.

119

5.1 Introduction
Today mechanically based sensors are ubiquitous, having a long history of
important applications in many diverse fields of science and technology. Among the most
responsive are sensors based on the acoustic vibratory modes of crystals,1,2 thin films,3
and

recently,

microelectromechanical

systems

(MEMS)4,5

and

nanoelectromechancial systems (NEMS).6,7 Two attributes of NEMS devices—minuscule
mass and high quality factor (Q)— provide them with unprecedented potential for mass
sensing. This is revealed in our analysis in chapter 3 and demonstrated by recently
achieved femtogram6 and attogram resolution.7 Attainment of zeptogram (1 zg=10-21 g)
sensitivity shown herein opens many new possibilities; among them is directly
“weighing” the inertial mass of individual, electrically neutral macromolecules.8 Such
sensitivity also enables the observation of extremely minute (statistical) mass fluctuations
that arise from the diffusion of atomic species upon the surfaces of NEMS devices—
processes that impose fundamental sensitivity limits upon nanoscale gas sensors (see
section 4.6). As an initial step into these applications, we perform mass sensing
experiments with gaseous species adsorbed on the NEMS surfaces at the zeptogram
scale.

5.2 Experimental Setup
All the experiments are done in situ within a cryogenically cooled, ultrahigh
vacuum apparatus with base pressure below 10-10 Torr. As shown in figure 5.1, a minute,
calibrated, highly controlled flux of Xe atoms or N2 molecules is delivered to the device
surface by a mechanically shuttered gas nozzle within the apparatus.9 The nozzle has an

120

orifice with a 100 μm diameter aperture, which is maintained at T=200 K by heating it
with ~1 W of power to prevent condensation of the gas within the orifice and its supply
line. Gas is delivered to this nozzle from a buffering chamber (volume VC =100 cm3 for
the N2 experiments, and 126 cm3 for the Xe experiments), in turn maintained at
temperature TC = 300 K.

Prior to the commencement of a run, this chamber is

pressurized with the species to be delivered, then sealed to allow escape only through the

nozzle. Thereafter, the rate of pressure decrease, P C , which is continuously monitored, is
proportional to the total adsorbate delivery rate from the gas nozzle to the NEMS sensor,
i.e., the number of incident atoms or molecules per unit time. The total number flux of

gas atoms or molecules out of the nozzle in steady state is given by N C = P C VC / k B TC .

Real-time mass sensing is enabled by the incorporation of NEMS device into a
VHF frequency modulation phase-locked loop (FM PLL), described in section 4.4. With
this measurement scheme, data are obtained from two separate sets of experiments
involving different NEMS resonators: a first device (hereafter, D133) with a fundamental
resonant frequency f0~133 MHz having dimensions: 2.3 μm (L) x 150 nm (w) x 70nm (t),
and a second (hereafter, D190) with f0~ 190 MHz and dimensions 2.3 μm (L) x 150 nm
(w) x 100 nm (t). Both are patterned from SiC epilayers10 and exhibit a quality factor of
Q =5000 in the temperature range of the present measurement.
For our experiment, the NEMS devices are maintained at high vacuum (~10-7
torr) at 300K for >1 day prior to mass accretion studies. The experiments are carried out
immediately after cryogenically cooling the devices in a background pressure below
~10-10 torr. Hence we assume the arriving species adsorb with unity sticking probability;

121

for our choices of Xe and N2 this is a reasonable assumption.11 The mass deposition rate
to the device is then

M D = m N D = mAD N C /(πLD ) ,

(5.1)

where m is the mass of adsorbed species ( m Xe =131 amu, m N 2 =28 amu), the factor AD/L2
is the solid angle of capture, AD is the surface area of the device exposed to the flux, and

LD is the distance between the device and nozzle.9 The weighting factor 1/π accounts for

the cosine distribution of the beam profile. For N2 experiment, N C = 2.25 x 1012

molecules/sec, LD = 2.37 cm, and AD = 3.45 x 10-13 m2, yielding M D =20.5 zg/sec. For

the Xe experiment, the setting are N C = 2.81 x 1012 atoms/sec, L D =1.86 cm, and

AD =3.45x10-13 m2. These values yield M D = 195 zg/sec.

122

NEMS
(heater mounted)

NEMS
shutter
nozzle

Figure 5.1. Experimental configuration. A gas nozzle with a 100 μm aperture provides

a controlled flux of atoms or molecules. The flux is gated by a mechanical shutter to
provide calibrated, pulsed mass accretions upon the NEMS device. The mass flux is
determined by direct measurements of the gas flow rate, in conjunction with effusivesource formulas for the molecular beam emanating from the nozzle.

123

5.3 Mass Sensing at Zeptogram Scale

We first demonstrate the real time, in situ, zeptogram-scale mass accretion on
D190, resulting from pulsed delivery of N2 molecules at T=37 K, as shown in figure 5.2.

With a mass deposition rate M D = 20.5 zg/sec, sequential openings of the shutter for 5
second intervals provides a series of 100 zg accretions. The resulting discretely stepped
frequency shifts tracked by the FM PLL confirm sequential, regular steps of mass
accretion (each ~100 zg, i.e., ~2000 N2 molecules).12 The mass sensitivity δM is set by
the

standard

deviation

δM = δf / ℜ = ( f − f 0 ) 2
Here ℜ = ∂f 0 / ∂M eff

1/ 2

frequency

fluctuations

/ℜ .

the

plateaus
(5.2)

is the mass responsivity of the device; M eff and f 0 are the

effective vibratory mass and resonant frequency of the device, respectively. The data of
figure 5.2 demonstrate δM =20 zg for the 1 sec averaging time employed.
The mass responsivities for the devices are directly determined from such pulsed
atom or molecule deposition measurements. Data are shown both for D190 (for
conditions described above) and for D133 in figure 5.3. We expose D133 to Xe with

mass deposition rate M D =195 zg/sec and opening shutter for 1 sec yields ~200 zg mass
accretion (or equivalently ~900 Xe atoms) per data point at T =46 K. Both devices
demonstrate unprecedented responsivities: ℜ , directly extracted from the slope of the
linear fit, at the level of roughly 1 Hz shift per zeptogram of accreted mass. More
precisely, we find ℜ D133 ≈ 0.96 Hz/zg and ℜ D190 ≈ 1.16 Hz/zg. These values are in
excellent agreement with the theoretical estimates from the expression ℜ ≈ f 0 / 2 M eff ,

124

which yields ~0.89 Hz/zg for D133 ( M eff ≈73 fg) and ~0.99 Hz/zg for D190 ( M eff ≈96
fg).8
Our highest mass sensitivity, at present, is demonstrated with D133 stabilized at

T= 4.2 K. Exceptionally small fractional frequency fluctuations δf / f 0 = ( f − f 0 ) 2

1/ 2

5×10-8 (50 ppb) are observed over a course of ~4000 sec interval with 1 sec averaging
time (right inset of figure 5.3). This demonstrates attainment of a mass sensitivity of

δM ~7 zg, equivalent to accretion of ~30 Xe atoms or, alternatively, of an individual 4
kDa macromolecule. Using M eff ≈73 fg, Q~5000, and dynamic range DR ~80 dB, such a
mass sensitivity is consistent with the estimated value 2.9 zg from the expression,

δM ~ (2 M eff / Q)10 − DR / 20 ,

(5.3)

as described by Ekinci et al.7 Our current dynamic range is determined, at the bottom end,
by the noise floor of the posttransducer readout amplifier of the NEMS device and, at the
top end, by the onset of nonlinearity arising from the Duffing instability for a doubly
clamped beam (see section 3.2). With our current experimental setup, we are able to track
mass accretions up to a total of ~2.3x106 Xe atoms on D190, with no observable change
in the quality factor (see section 4.6). This confirms a remarkably large mass dynamic
range from a few kDa (or several zeptogram sensitivity) up to ~100 MDa range,
corresponding to almost femtogram-scale accretions.

125

Frequency Shift (Hz)

-200

~100 zg

-400
-600
-800
-1000

100 150 200 250 300 350
Time (sec)

Figure 5.2. Real time zeptogram-scale mass-sensing experiment. Sequential mass

depositions are executed in situ upon the 190 MHz device within a cryogenic UHV
apparatus. The resulting frequency shift of the NEMS device is tracked in real time by a
very high frequency (VHF) phase-locked loop. Each step in the data corresponds to a
~100 zg mass accretion (~2000 N2 molecules) resulting from opening the mechanical
shutter for 5 sec. The root-mean-square frequency fluctuations of the system correspond
to a mass sensitivity of δM = 20 zg for the 1 sec averaging time employed.

126

100

δ m (zg)

Frequency Shift (Hz)

-500

-1000
0.1

-1500

2000

4000

Time (s)

-2000

133 MHz
190 MHz

-2500
-3000

1000

2000

3000

4000

Mass (zeptograms)
Figure 5.3. Mass responsivities of nanomechanical devices. The mass responsivities

(resonant frequency shifts versus accreted mass) are measured for two VHF NEMS
devices (operating at 133 MHz and 190 MHz). Xe atoms are accreted at T=46 K upon
the 133 MHz device with ~200 zg mass increments per data point (purple). N2 molecules
are accreted at T=37 K upon the 190 MHz device with ~100 zg mass increments per data
point (blue). The slopes of the mass loading curves directly exhibit the unprecedented
mass responsivity on the order 1 Hz per zeptogram. (right inset) Mass sensitivity. The
“mass noise floor” for the 133 MHz device, which originates from its frequencyfluctuation noise, is measured with 1 sec averaging time over the course of ~4000 sec
while it is temperature stabilized at 4.2 K with zero applied flux. The average root-meansquare value (red dashed line), reflects the attainment of ~7 zg (i.e., ~4 kDa) mass
sensitivity, the equivalent of ~30 Xe atoms.

127

5.4 Conclusion

We demonstrate NEMS mass sensing at the zeptogram scale. The agreement
between predicted and experimentally observed values for both ℜ and δM confirms our
analyses in chapter 3 and validates their use in projecting a path toward single-Dalton
mass sensitivity.8 Attainment of this goal is possible, for example, with a device having a
fundamental resonant frequency of 1 GHz, vibratory mass of Meff=1x10-16 g, and

Q=10,000, using a transduction-readout system providing DR=80 dB. These are realistic
parameters for next generation NEMS.13 Huang, et al. recently attained NEMS vibrating
in fundamental mode at microwave frequencies.13 In conjunction with the recent
development of techniques for improved Q,14 and the advances in frequency-shift readout
in the tens of ppb range, it is clear that NEMS sensing at the level of ~1 Da will soon be
within reach. Attainment of NEMS mass sensors with single-Dalton sensitivity will make
feasible the detection of individual, intact, electrically neutral macromolecules with
masses ranging well into the hundreds of MDa range. This is an exciting prospect —
when realized it will blur the traditional distinction between inertial mass sensing and
mass spectrometry. We anticipate that it will also open intriguing possibilities in atomic
physics and life science.15,16

128

References

D. S. Ballantine et al. Acoustic Wave Sensors (San Diego, Academic Press, 1997).

C. Lu Application of Piezoelectric Quartz Crystal Microbalance (London,
Elsevier, 1984).

M. Thompson and D. C. Stone Surface-Launched Acoustic Wave Sensors:
Chemical Sensing and Thin Film Characterization (New York, John Wiley and
Sons, 1997).

J. Thundat, E. A. Wachter, S. L. Sharp, and R. J. Warmack Detection of mercury
vapor using resonating microcantilevers. Appl. Phys. Lett. 66, 1695–1697 (1995).

Z. J. Davis, G. Abadal, O. Kuhn, O. Hansen, F. Grey, and A. Boisen Fabrication
and characterization of nanoresonating devices for mass detection. J. Vac. Sci.
Technol. B 18, 612-616 (2000).

N. V. Lavrik and P. G. Datskos Femtogram mass detection using photothermally
actuated nanomechanical resonators. Appl. Phys. Lett 82, 2697 (2003).

K. L. Ekinci, X. M. H. Huang, and M. L. Roukes Ultrasensitive
nanoelectromechanical mass sensing. Appl. Phy. Lett. 84, 4469 (2004).

K. L. Ekinci, Y. T. Yang, and M. L. Roukes Ultimate limits to inertial mass
detection based upon nanoelectromechanical systems. J. Appl. Phys. 95, 2682
(2004).

H. Pauly and G. Scoles (editor) Atomic and Molecular Beam Methods (New
York, Oxford University Press, 1988).

10.

Y. T. Yang et al. Monocrystalline silicon carbide nanoelectromechancial systems.
Appl. Phys. Lett. 78, 162-164 (2001).

11.

H. J. Kreuzer and Z. W. Gortel Physisorption Kinetics (Springer-Verlag, New
York, 1986). At low coverage, unity sticking probability are observed for Xe on
W(100) at T= 65 K. Xe on Ni(100) at T=30 K and N2 on Ni(110) T=87 K. We
expect the above to be representative materials and conditions, so that cryogenic
adsorption upon the NEMS device will behave similarly in our case.

12.

We have also verified that the arriving species provide negligible thermal
perturbation upon the device. This is accomplished by high-resolution in situ
resistance thermometry upon the metallic displacement-transducer electrode, and
comparing shutter-open and shutter-closed conditions. The kinetic energy of the
arriving species negligibly perturbs the device.

129

13.

X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes Nanodevices
motion at microwave frequencies. Nature 421, 496 (2003).

14.

X. M. H. Huang, X. L. Feng, C. A. Zorman, M. Mehregany, and M. L. Roukes
Free free beam silicon carbide nanomechanical resonators. New J. Phys. 7, 247
(2005).

15.

W. Hansel, P. Hommelhoff, T. W. Hansh, and J. Reichel Bose-Einstein
condensation on microelectronic chips. Nature 413, 498-500 (2001).

16.

R. Aebersold and M. Mann Mass spectrometry-based proteomics. Nature 422,
198-207 (2003).

130

Chapter 6*
Monocrystalline Silicon Carbide
Nanoelectromechanical Systems

SiC is an extremely promising material for nanoelectromechanical systems given its
large Young’s modulus and robust surface properties. We have patterned
nanometer scale electromechanical resonators from single-crystal 3C-SiC layers
grown epitaxially upon Si substrates. A surface nanomachining process is described
that involves electron beam lithography followed by dry anisotropic and selective
electron cyclotron resonance plasma etching steps. Measurements on a
representative family of the resulting devices demonstrate that, for a given
geometry, nanometer-scale SiC resonators are capable of yielding substantially
higher frequencies than GaAs and Si resonators.

This section has been published as: Y. T. Yang, K. L. Ekinci, X. M. H. Huang, L. M. Schiavone, M. L.
Roukes, C. A. Zorman, and M. Mehregany, Appl. Phys. Lett. 78, 162-164 (2001).

131

6.1 Introduction
Silicon carbide is an important semiconductor for high temperature electronics
due to its large band gap, high breakdown field, and high thermal conductivity. Its
excellent mechanical and chemical properties have also made this material a natural
candidate for microsensor and microactuator applications in microelectromechanical
systems (MEMS).1,2
Recently, there has been a great deal of interest in the fabrication and
measurement of semiconductor devices with fundamental mechanical resonance
frequencies reaching into the microwave bands.3 Among technological applications
envisioned for these nanoelectromechanical systems (NEMS) are ultrafast, highresolution actuators and sensors, and high frequency signal processing components and
systems. From the point of view of fundamental science, NEMS also offer intriguing
potential for accessing regimes of quantum phenomena and for sensing at the quantum
limit.4
SiC is an excellent material for high frequency NEMS for two important reasons.
First, the ratio of its Young’s modulus, E, to mass density, ρ, is significantly higher than
for other semiconducting materials commonly used for electromechanical devices, e.g.,
Si and GaAs. Flexural mechanical resonance frequencies for beams directly depend upon
the ratio

E / ρ . The goal of attaining extremely high fundamental resonance

frequencies in NEMS, while simultaneously preserving small force constants necessary
for high sensitivity, requires pushing against the ultimate resolution limits of lithography
and nanofabrication processes. SiC, given its larger E / ρ , yields devices that operate at
significantly higher frequencies for a given geometry, than otherwise possible using
132

conventional materials. Second, SiC possesses excellent chemical stability.3 This makes
surface treatments an option for higher quality factors (Q factor) of resonance. It has been
argued that for NEMS the Q factor is governed by surface defects and depends on the
device surface-to-volume ratio.2
Micron-scale SiC MEMS structures have been fabricated using both bulk and
surface micromachining techniques. Bulk micromachined 3C-SiC diaphragms, cantilever
beams, and torsional structures have been fabricated directly on Si substrates using a
combination of 3C-SiC growth processes and conventional Si bulk micromachining
techniques in aqueous KOH and TMAH solutions.5,6 Surface micromachined SiC devices
have primarily been fabricated from polycrystalline 3C-SiC (poly-SiC) thin films
deposited directly onto silicon dioxide sacrificial layers, patterned using reactive ion
etching, and released by timed etching in aqueous hydrofluoric acid solutions.8 Single
crystal 3C-SiC surface micromachined structures have been fabricated in a similar way
from 3C-SiC-on-SiO2 substrates created using wafer bonding techniques.9 We have
developed an alternative approach for nanometer-scale single crystal, 3C-SiC layers that
is not based upon wet chemical etching or wafer bonding. Especially noteworthy is that
our final suspension step in the surface nanomachining process is performed by using a
dry etch process. This avoids potential damage due to surface tension encountered in wet
etch processes, and circumvents the need for critical point drying when defining large,
mechanically compliant devices. We first describe the method we developed for
fabrication of suspended SiC structures, then demonstrate the high frequency
performance attained from doubly clamped beams read out using magnetomotive
detection.

133

6.2 Device Fabrication and Measurement Results
The starting material for device fabrication is a 259 nm thick single crystalline
3C-SiC film heteroepitaxially grown on a 100 mm diameter (100) Si wafer. 3C-SiC
epitaxy is performed in a rf induction-heated reactor using a two-step, carbonizationbased atmospheric pressure chemical vapor deposition (APCVD) process detailed
elsewhere.10 Silane and propane are used as process gases and hydrogen is used as the
carrier gas. Epitaxial growth is performed at a susceptor temperature of about 1330 °C.
3C-SiC films grown using this process have a uniform (100) orientation across each
wafer, as indicated by x-ray diffraction. Transmission electron microscopy and selective
area diffraction analysis indicates that the films are single crystalline. The microstructure
is typical of epitaxial 3C-SiC films grown on Si substrates, with the largest density of
defects found near the SiC/Si interface, which decreases with increasing film thickness. A
unique property of these films is that the 3C-SiC/Si interface is absent of voids, a
characteristic not commonly reported for 3C-SiC films grown by APCVD.
Fabrication begins by defining large area contact pads by optical lithography. A
60 nm thick layer of Cr is then evaporated and, subsequently, standard lift-off is carried
out with acetone. Samples are then coated with a bilayer polymethylmethacrylate
(PMMA) resist prior to patterning by electron beam lithography. After resist exposure
and development, 30–60 nm of Cr is evaporated on the samples, followed by lift-off in
acetone. The pattern in the Cr metal mask is then transferred to the 3C-SiC beneath it by
anisotropic electron cyclotron resonance (ECR) plasma etching. We use a plasma of NF3,
O2, and Ar at a pressure of 3 mTorr with respective flow rates of 10, 5, 10 sccm, and a

134

microwave power of 300 W. The acceleration dc bias is 250 V. The etch rate under these
conditions is ~65 nm/min.
The vertically etched structures are then released by controlled local etching of
the Si substrate using a selective isotropic ECR etch for Si. We use a plasma of NF3 and
Ar at a pressure of 3 mTorr, both flowing at 25 sccm, with a microwave power 300 W,
and a dc bias of 100 V. We find that NF3 and Ar alone do not etch SiC at a noticeable rate
under these conditions. The horizontal and vertical etch rates of Si are ~300 nm/min.
These consistent etch rates enable us to achieve a significant level of control of the
undercut in the clamp area of the structures. The distance between the suspended
structure and the substrate can be controlled to within 100 nm.
After the structures are suspended, the Cr etch mask is removed either by ECR
etching in an Ar plasma or by a wet Cr photomask etchant (perchloric acid and ceric
ammonium nitrate). The chemical stability and the mechanical robustness of the
structures allow us to perform subsequent lithographic fabrication steps for the requisite
metallization (for magnetomotive transduction) step on the released structures.
Suspended samples are again coated with bilayer PMMA and after an alignment step,
patterned by electron beam lithography to define the desired electrodes. The electrode
structures are completed by thermal evaporation of 5 nm thick Cr and 40 nm thick Au
films, followed by standard lift-off. Finally, another photolithography step, followed by
evaporation of 5 nm Cr and 200 nm Au and conventional lift-off, is performed to define
large contact pads for wire bonding. Two examples of completed structures, each
containing a family of doubly clamped SiC beams of various aspect ratios, are shown in
figure 6.1.

135

Figure 6.1. SEM picture of doubly clamped SiC beams. Doubly clamped SiC beams
patterned from a 259 nm thick epilayer. (left) Top view of a family of 150 nm wide
beams, having lengths from 2 to 8 μm. (right) Side view of a family of 600 nm wide
beams, with lengths ranging from 8 to 17 μm.

136

We have measured the fundamental resonance frequencies of both the in-plane
and out-of-plane vibrational modes for a family of doubly clamped SiC beams, with
rectangular cross section and different aspect ratios (length/width). Samples were glued
into a chip carrier and electrical connections were provided by Al wirebonds.
Electromechanical characteristics were measured using the magnetomotive detection
technique11 from 4.2 to 295 K, in a superconducting solenoid within a variable
temperature cryostat. The measured fundamental frequencies in this study ranged from
6.8 to 134 MHz. The quality factors, extracted from the fundamental mode resonances for
each resonator, range from 103representative beam with dimensions 8 μm (length) × 600 nm (width) × 259 nm
(thickness). This particular device yields an in-plane resonant frequency of 71.91 MHz
and a Q=4000 at 20 K. Quality factors at room temperature were typically a factor of 4–5
smaller than values obtained at low temperature.

137

4.0

Amplitude (μV)

3.5
3.0

5.5 T
5T
4T
3T
2T
1T
0T

2.5
2.0
1.5
1.0
0.5
0.0
71.80

71.85

71.90

71.95

72.00

72.05

Frequency (MHz)
Figure 6.2. Representative data of mechanical resonance. A SiC doubly clamped
beam resonating at 71.91 MHz, with quality factor Q=4000. The family of resonance
curves are taken at various magnetic fields; the inset shows the characteristic B 2
dependence expected from magnetomotive detection. For clarity of presentation here the
data are normalized to response at zero magnetic field, with the electrode’s dc
magnetoresistance shift subtracted from the data; these provide an approximate means for
separating the electromechanical response from that of the passive measurement
circuitry.

138

We now demonstrate the benefits of SiC for NEMS, by directly comparing
frequencies attainable for structures of similar geometry made with SiC, Si, and GaAs.
The fundamental resonance frequency, f 0 , of a doubly clamped beam of length, L , and
thickness, t, varies linearly with the geometric factor t/L2 according to the simple relation

f 0 = 1.03

E t
ρ L2

(6.1)

where E is the Young’s modulus and ρ is the mass density. In our devices the resonant
response is not so simple, as the added mass and stiffness of the metallic electrode
modify the resonant frequency of the device. This effect becomes particularly significant
as the beam size shrinks. To separate the primary dependence upon the structural material
from secondary effects due to electrode loading and stiffness, we employ a simple model
for the composite vibrating beam.12 In general, for a beam comprised of two layers of
different materials the resonance equation is modified to become
f0 =

η ⎛ E1 I 1 + E 2 I 2 ⎞
⎟.
L2 ⎜⎝ ρ1 A1 + ρ 2 A2 ⎟⎠

(6.2)

Here the indices 1 and 2 refer to the geometric and material properties of the structural
and electrode layers, respectively. The constant η depends upon mode number and
boundary conditions; for the fundamental mode of a doubly clamped beam η =3.57.
Assuming the correction due to the electrode layer (layer 2) is small, we can define a
correction factor K, to allow direct comparison with the expression for homogeneous
beam
η ⎛EI
f 0 = 2 ⎜⎜ 1 10 K ⎟⎟
L ⎝ ρ1 A10 ⎠

1/ 2

, where K =

E1 I 1 + E 2 I 2
E1 I 10

ρ A
1+ 2 2
ρ1 A1

(6.3)

139

Frequency (MHz)

200
100

10
SiC
Si
GaAs

-4

-3

-2

Effective Geometric Factor, [ t / L ]eff (μm )

-1

Figure 6.3. Frequency versus effective geometric factor for three families of doubly
clamped beams made from single-crystal SiC, Si, and GaAs. All devices are patterned

to have the long axis of the device along <100>. Ordinate are normalized to remove the
effect of additional stiffness and mass loading from electrode metallization. The solid
lines are least squares fits assuming unity slope, and yield values of the parameter
v = E / ρ that closely match expected values.

140

In this expression, I10 is the moment calculated in the absence of the second layer.
The correction factor K can then be used to obtain a value for the effective geometric
factor, [t / L2 ]eff , for the measured frequency.13 Further nonlinear correction terms, of

order higher than [t / L2 ]eff , are expected to appear if the beams are under significant
tensile or compressive stress. The linear trend of our data, however, indicates that internal
stress corrections to the frequency are small.
In figure 6.3, we display the measured resonance frequencies as a function of
[t / L2 ]eff for beams made of three different materials: GaAs, Si and SiC.14 The lines in

this logarithmic plot represent least squares fits to the data assuming unity slope. From
these we can deduce the effective values of the parameter, v = E / ρ , which is similar
(but not identical) to the velocity of sound for the three materials.15 The numerical values
obtained by this process are: v(SiC ) =1.5×104 m/s,

v(Si ) =8.4×103 m/s, and

v(GaAs) =4.4 ×103 m/s. These are quite close to values calculated from data found in the
literature: v(SiC ) =1.2×104 m/s,16 v(Si ) =7.5×103 m/s,17 and v(GaAs) =4.0×103 m/s,18
respectively. The small discrepancies are consistent with our uncertainties in determining
both the exact device geometries and the precise perturbation of the mechanical response
arising from the metallic electrodes. Nonetheless, SiC very clearly exhibits the highest
E / ρ ratio.17

6.3 Conclusion

In conclusion, we report a simple method for fabricating nanomechanical devices
from single-crystal 3C-SiC materials. We demonstrate patterning mechanical resonators
using a single metal mask, and just two steps of ECR etching. Our results illustrate that
141

SiC is an ideal semiconductor with great promise for device applications requiring high
frequency mechanical response.

142

References

M. Mehregany, C. A. Zorman, N. Rajan, and C. H. Wu Silicon carbide MEMS for
harsh environments. Proc. IEEE 86, 1594 (1998).

M. L. Roukes Nanoelectromechanical systems. Technical Digest of the 2000
Solid-State Sensor and Actuator Workshop, Hilton Head Island, South Carolina
(June 4–8 2000) 367–376 (2000).

X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes Nanodevices
motion at microwave frequencies. Nature 421, 496 (2003).

M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab Approaching the quantum
limit of a nanomechanical resonator. Science 304, 74 (2004).

P. A. Ivanov and V. E. Chelnokov Recent developments in SiC single-crystal
electronics. Semicond. Sci. Technol. 7, 863 (1992).

L. U. Tong and M. Mehregany Mechanical-properties of 3C-Silicon Carbide.
Appl. Phys. Lett. 60, 2992 (1992).

C. Serre, A. Perez-Rodriguez, A. Romano-Rodriguez, J. R. Morante, J. Esteve,
and M. C. Acero Test microstructures for measurement of SiC thin firm
mechanical properties. J. Micromech. Microeng. 9, 190 (1999).

A. J. Fleischman, X. Wei, C. A. Zorman, and M. Mehregany Surface
micromachining of polycrystalline SiC deposited on SiO2 by APCVD. Proc. of
the IEEE International Conference on Silicon Carbide, III-Nitrides, and Related
Materials (1997) 885–888 (1998).

S. Stefanescu, A. A. Yasseen, C. A. Zorman, and M. Mehregany Surface
micromachined lateral resonant structures fabricated from single crystal 3C-SiC
films, Proceeding of the 10th International Conference on Solid State Sensors and
Actuators, Sendai, Japan (June 7–10 1999), 194–198 (1999).

10.

C. A. Zorman, A. J. Fleischman, A. S. Dewa, M. Mehregany, C. Jacob, S.
Nishino, and P. Pirouz Epitaxial-growth of 3C-SiC films on 4 inch diam. (100)
silicon-wafers by atmospheric-pressure chemical-vapor-deposition. J. Appl. Phys.
78, 5136 (1995).

11.

A. N. Cleland and M. L. Roukes Fabrication of high frequency nanometer scale
mechanical resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653 (1996).

12.

K. E. Petersen and C. R. Guarnieri Young's modulus measurements of thin-films
using micromechanics. J. Appl. Phys. 50, 676 (1979).

143

13.

The correction factor K primarily reflects mass loading from the metallic
electrode. Using values from the literature for Young's modulus of the electrode
materials we deduce that the additional stiffness introduced is completely
negligible.

14.

Electrodes were composed of either Au or Al, with typical thickness ranging from
50 to 80 nm.

15.

The quantity

E<100> / ρ is strictly equal to neither the longitudinal sound

velocity, c11 / ρ , nor the transverse sound velocity, c44 / ρ for propagation
along <100> direction of cubic crystal. Here the cs are elements of the elastic
tensor and E<100> = (c11 − c12 )(c11 + 2c12 ) /(c11 + c12 ) for cubic crystal. See, e.g. B.
A. Auld, Acoustic Fields and Waves in Solids, 2nd edition (Robert E. Krieger,
Malabar, 1990)
16.

W. R. L. Lambrecht, B. Segall, M. Methfessel, and M. Vanschilfgaarde
Calculated elastic-constants and deformation potentials of cubic SiC. Phys. Rev. B
44, 3685 (1991).

17.

J. J. Hall Electronic effects in elastic constants of n-Type silicon. Phys. Rev. 161,
756 (1967).

18.

R. I. Cottam and G. A. Saunders Elastic-constants of GaAs from 2 K to 320 K. J.
Phys. C: Solid State 6, 2105 (1973).

144

Chapter 7*
Balanced Electronic Detection of
Displacement in
Nanoelectromechanical Systems

We describe a broadband radio frequency balanced bridge technique for electronic
detection of displacement in nanoelectromechanical systems (NEMS). With its twoport actuation-detection configuration, this approach generates a backgroundnulled electromotive force in a dc magnetic field that is proportional to the
displacement of the NEMS resonator. We demonstrate the effectiveness of the
technique by detecting small impedance changes originating from NEMS
electromechanical resonances that are accompanied by large static background
impedances at very high frequencies. This technique allows the study of important
experimental systems such as doped semiconductor NEMS and may provide
benefits to other high frequency displacement transduction circuits.
© 2002 American Institute of Physics. [DOI: 10.1063/1.1507833]

This section has been published as: K. K. L. Ekinci, Y. T. Yang, X. M. H. Huang, and M. L. Roukes,
Appl. Phys. Lett. 78 162 (2002).

145

7.1 Introduction
The recent efforts to scale microelectromechanical systems (MEMS) down to the
submicron domain1 have opened up an active research field. The resulting
nanoelectromechanical systems (NEMS) with fundamental mechanical resonance
frequencies reaching into the microwave bands are suitable for a number of important
technological applications. Experimentally, they offer potential for accessing interesting
phonon mediated processes and the quantum behavior of mesoscopic mechanical
systems.
Among the most needed elements for developing NEMS based technologies—as
well as for accessing interesting experimental regimes—are broadband, on-chip
transduction methods sensitive to subnanometer displacements. While displacement
detection at the scale of MEMS has been successfully realized using magnetic,2
electrostatic3,4 and piezoresistive5 transducers through electronic coupling, most of these
techniques become insensitive at the submicron scales.

7.2 Circuit Schemes and Measurement Results
An on-chip displacement transduction scheme that scales well into the NEMS
domain and offers direct electronic coupling to the NEMS displacement is
magnetomotive detection.6,7 Magnetomotive reflection measurements as shown
schematically8 in figure 7.1(a) have been used extensively.6,7,9 Here, the NEMS resonator
is modeled as a parallel RLC network with a mechanical impedance, Z m (ω ) , a twoterminal dc coupling resistance, Re , and mechanical resonance frequency, ω0 . When

146

(a)

(b)
D1

Re+ΔR

Vo
Re

Vin

0º

RS
PS

Vin

180º

NEMS

D2
D1

RO
DC

D2
PS

Vin

(C)

Figure 7.1. Schematic diagrams for the magnetomotive reflection measurement and
bridge measurement (a) Schematic diagram for the magnetomotive reflection
mesurement. In both reflection and bridges measurements, a network analyzer (NA)
supplies the drive voltage, Vin. In reflection measurement, a directional coupler (DC) is
implemented to access the reflected signal from the device. (b) Schematic diagram for
the magnetomotive bridge measurement. Vin is split into two out-of-phase components
by a power splitter (PS) before it is applied to ports D1 and D2. (c) Scanning electron
micrograph of a representative bridge device, from an epitaxially grown wafer with 50
nm thick n + GaAs and 100 nmGaAs structure layer on top. The doubly clamped beams
with dimensions of 8 μm(L) × 150 nm(w) × 500 nm(t) at the two arms of the bridge have
in plane fundamental flexural mechanical resonances at ~35 MHz. D1, D2, and RO ports
on the device are as shown.

147

driven at ω by a source with impedance Rs , the voltage on the load, RL , can be
approximated as
V0 (ω ) = Vin (ω )

Re + Z m (ω )
R + Z m (ω )
≅ Vin (ω ) e
RL + ( Re + Z m (ω ))
RL + Re

(7.1)

Here, RL = RS = 50Ω . We have made the approximation that Re >> Z m (ω ) , as is the
case in most experimental systems. Apparently, the measured electromotive force (EMF)
due to the NEMS displacement proportional to Z m (ω ) is embedded in a background
close to the drive voltage amplitude, V0 ~ Vin − 20 log Re /( RL + Re ) dB.10 This facilitates
the definition of a useful parameter at ω = ω0 , the detection efficiency, S/B, as the ratio of
the signal voltage to the background. For the reflective, one-port magnetomotive
measurement

figure

7.1(a),

S / B = Z m (ω 0 ) / Re = Rm / Re ,

indicating

some

shortcomings. First, detection of the EMF becomes extremely challenging, when
Re << Rm , i.e., in unmetallized NEMS devices or metallized high frequency NEMS
(small Rm ). Second, the voltage background prohibits the use of the full dynamic range
of the detection electronics. In addition to the balanced bridge detection here, we describe
two-port schem to improve the detection efficiency, i.e., S/B ratio.11
The balanced circuit shown in figure 7.1(b) with a NEMS resonator on one side of
the bridge and a matching resistor of resistance, R = Re + ΔR on the other side, is
designed to improve S/B. The voltage, V0 (ω ) at the readout (RO) port is nulled for

ω ≠ ω0 , by applying two 180° out of phase voltages to the Drive 1 (D1) and Drive 2 (D2)
ports in the circuit. We have found that the circuit can be balanced with exquisite
sensitivity, by fabricating two identical doubly clamped beam resonators on either side of
148

the balance point (RO), instead of a resonator and a matching resistor, as shown in figure
7.1(c). In such devices, we almost always obtained two well-separated mechanical
resonances, one from each beam resonator, with ω2 − ω1 >> ω1 / Q where ω i and Q are
the resonance frequency and the quality factor of resonance of the resonators (i=1,2) (see
figure 7.3). This indicates that in the vicinity of either mechanical resonance, the system
is well described by the mechanical resonator-matching resistor model of figure 7.1(b).
We attribute this behavior to the high Q factors ( Q ≥ 10 3 ) and the extreme sensitivity of
the resonance frequencies to local variations of parameters during the fabrication process.
First, to clearly assess the improvements, we compared reflection and balanced
bridge measurements of the fundamental flexural resonances of doubly clamped beams
patterned from n + (B-doped) Si as well as from n + (Si-doped) GaAs. Electronic
detection of mechanical resonances of these types of NEMS resonators have proven to be
Challenging,12 since for these systems Re ≥ 2kΩ and Rm ≤ Re . Nonetheless, with the
bridge technique we have detected fundamental flexural resonances in the 10 MHz<85 MHz range for n + Si resonators and in the 7 MHz beams. In all our measurements, the paradigm that Rm << Re remained true as Rm = 10 Ω
and 2 kΩ < Re < 20 kΩ. Here, we focus on our results from n + Si beams. These were
fabricated from a B-doped Si on insulator wafer, with Si layer and buried oxide layer
thicknesses of 350 and 400 nm, respectively. The doping was done at 950 °C. The dopant
concentration was estimated as N d ≈ 6x1019 cm-3 from the sample sheet resistance,
R□=60 Ω.13 The fabrication of the actual devices involved optical lithography, electron
beam lithography, and lift-off steps followed by anisotropic electron cyclotron resonance
149

plasma and selective HF wet etches.7,9,12 The electromechanical response of the bridge
was measured in a magnetic field generated by a superconducting solenoid. Figure 7.2(a)
shows the response of a device with dimensions 15 μm(L) × 500 nm(w) × 350 nm(t) and
with Re = 2.14 kΩ, measured in the reflection (upper curves) And bridge configurations
for several magnetic field strengths. The device has an in plane flexural resonance at
25.598 MHz with Q = 3x104 at T = 20 K. With ΔR ≈ 10 Ω a background reduction of a
factor of Re / ΔR = 200 was obtained in the bridge measurements (see analysis below).
Figure 7.2(b) shows a measurement of the broadband transfer functions for both
configurations for comparable drives at zero magnetic field. Notice the dynamic
background reduction in the relevant frequency range.

150

(a)
88.5

Amplitude (μV)

88.0

0.4
0.3
0.2
6T

0.1
0.0
25.590

25.595

25.600

25.605

Frequency (MHz)

(b)
Reflection

|V0/Vin| (dB)

-20
Bridge

-40

-60

100

150

200

Frequency (MHz)

Figure 7.2 Data from a doubly clamped n+ Si beam. (a) Mechanical resoanance. The
mechanical resonance at 25.598 MHz with a Q~3x104 from a doubly clamped, n + Si
beam is measured in reflection (upper curves) and in bridge (lower curves) configurations
for magnetic field strengths of B=0,2,4,6 T. The drive voltages are equal. The background
is reduced by a factor of ~200 in the bridge measurements. The phase of the resonance in
the bridge measurements can be shifted 180° with respect to the drive signal (see
Fig.7.1). (b) The amplitude of the broadband transfer functions. The broadband
transfer function H B (ω ) = V0 (ω ) / Vin (ω ) for reflection (upper curve) and bridge (lower
curve) configurations. The data indicate a background reduction of at least 20 dB and
capacitive coupling between the actuation–detection ports in the bridge circuit.
151

Bridge measurements also provided benefits in the detection of electromechanical
resonances from metallized VHF NEMS. These systems generally possess high Re and
Rm diminishes quickly as the resonance frequencies increase. Here, we present from our
measurements on doubly clamped SiC beams embedded within the bridge configuration.
These beams were fabricated with top metallization layers using a process described in
detail.9 For such beams with Re = 100 Ω and Rm ≤ 1 Ω, we were able to detect mechanical
flexural resonances deep into the VHF band. Figure 7.3(a) depicts a data trace of the in
plane flexural mechanical resonances of two 2 μm (L) × 150 nm (w) × 80 nm (t) doubly
clamped SiC beams. Two well-separated resonances are extremely prominent at 198.00
and 199.45 MHz, respectively, with Q=103 at T= 4.2 K. The broadband response from
the same device is plotted in figure 7.3(b). A reflection measurement in the vicinity of the
mechanical resonance frequency of this system would give rise to an estimated
background on the order, V0 / Vin =-20dB,10 making the detection of the resonance
extremely challenging.

152

(a)
-103
-104

|S21| (dB)

-105

-106
-107
-108
8T
-109
197.0 197.5 198.0 198.5 199.0 199.5 200.0 200.5
Frequency (MHz)

(b)
-90

|S21| (dB)

-100

-110

-120
0.0

0.2

0.4

0.6

0.8

1.0

1.2

Frequency (GHz)

Figure 7.3. Narrow band and broadband transfer function (S21) amplitudes from
metallized SiC beams in bridge configuration. (a) The narrow band response. The
narrowband response is measured for different magnetic field strengths of B=2, 4, 6, 8 T
and shows two well-separated resonances at 198.00 and 199.45 MHz, respectively, with
Q=103. (b) The broadband response. The broadband response at B=0 T shows the
significant background nulling attainable in bridge measurements. We estimate that a
reflection measurement on this system would produce V0 / Vin =20dB for ω ≈ ω 0 .

153

Figure 7.1(b) depicts our analysis of the bridge circuit. The voltage at point RO in
the circuit can be determined as14
V0 (ω ) = −

Vin (ω )[ΔR + Z m (ω )]
V (ω )
= − in'
[ ΔR + Z m (ω )] ,
( ΔR + Z m (ω ))(1 + Re / R L ) + Re ( 2 + Re / RL )
Z eq (ω )

(7.2)

in analogy to equation (7.1). At ω = ω0 , S / B = Rm / ΔR . Given that ΔR is small, the
background is suppressed by a factor of order Re / ΔR , as compared to the one-port case
as shown in figure 7.2(a). At higher frequencies, however, the circuit model becomes
imprecise as is evident from the measurements of the transfer function. Capacitive
coupling becomes dominant between D1, D2, and RO ports as displayed in figure 7.2(b),
and this acts to reduce the overall effectiveness of the technique. With careful design of
the circuit layout and the bonding pads, such problems can be minimized. Even further
signal improvements can be obtained by addressing the significant impedance mismatch,
Re ≥ RL , between the output impedance, Re , and the amplifier input impedance, RL . In
the measurements displayed in figure 7.2(a), this mismatch caused a signal attenuation
estimated to be of order ~40 dB.
Our measurements on doped NEMS offer insight into energy dissipation
mechanisms in NEMS, especially those arising from surfaces and surface adsorbates. In
the frequency range investigated, 10 MHz < f0 < 85 MHz, the measured Q factors of
2.2x104 < Q < 8x104 in n + Si beams is a factor of 2–5 higher than those obtained from
metallized beams.15 Both metallization layers16 and impurity dopants3 can make an
appreciable contribution to the energy dissipation. Our measurements on NEMS seem to
confirm that metallization overlayers can significantly reduce Q factor. The high Q
factors attained and the metal free surfaces make doped NEMS excellent tools for the
154

investigation of small energy dissipation changes due to surface adsorbates and defects.
In fact, efficient in situ resistive heating in doped beams through Re has been shown to
facilitate thermal annealing17 and desorption of surface adsorbates—yielding even higher
quality factors.

7.3 Conclusion
In conclusion, we have developed a broadband, balanced radio frequency bridge
technique for detection of small NEMS displacements. This technique may prove useful
for other high frequency high impedance applications such as piezoresistive displacement
detection. The technique, with its advantages, has enabled electronic measurements of
NEMS resonances otherwise essentially unmeasurable.

155

References

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(2001).

D. S. Greywall, B. Yurke, P. A. Busch, A. N. Pargellis, and R. L. Willett Evading
amplifier noise in nonlinear oscillators. Phys. Rev. Lett. 72, 2992 (1994).

R. E. Mihailovich and J. M. Parpia Low-temperature mechanical-properties of
boron-doped silicon. Phys. Rev. Lett. 68, 3052 (1992).

W. C. Tang, T. C. H. Nguyen, M. W. Judy, and R. T. Howe Electrostatic-comb
drive of lateral polysilicon resonators. Sens. Actuator A 21, 328 (1990).

M. Tortonese, R. C. Barrett, and C. F. Quate Atomic resolution with an atomic
force microscope using piezoresistive detection. Appl. Phys. Lett. 62, 834 (1993).

A. N. Cleland and M. L. Roukes Fabrication of high frequency nanometer scale
mechanical resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653 (1996).

A. N. Cleland and M. L. Roukes External control of dissipation in a nanometerscale radiofrequency mechanical resonator. Sens. Actuator A 72, 256 (1999).

To simplify, the length of the transmission line, lt , between the NEMS and the
measurement point has been set to l t ≈ λ / 2 where λ is the drive wavelength.
Also, the reflection coefficient, Γ , from the NEMS, defined as the ratio of the
amplitudes of reflected to incident voltages, is taken as unity. Experimentally, l is
readily adjustable and Γ closed to unity with Re close to 100 Ω → 1 kΩ.

Y. T. Yang, K. L. Ekinci, X. M. H. Huang, L. M. Schiavone, M. L. Roukes, C. A.
Zorman, and M. Mehregany Monocrystalline silicon carbide
nanoelectromechanical systems. Appl. Phys. Lett. 78, 162 (2001).

10.

When Γ≠1, V0 ≅ ΓVin ( Re + Z (ω )) /( RL + Re ) , giving a correction to the
background on the order of − 20 log Γ dB.

11.

See chapter 4.

12.

L. Pescini, A. Tilke, R. H. Blick, H. Lorenz, J. P. Kotthaus, W. Eberhardt, and D.
Kern Suspending highly doped silicon-on-insulator wires for applications in
nanomechanics. Nanotechnology 10, 418 (1999).

13.

S. M. Sze Physics of Semiconductor Devices (New York, Wiley, 1981).

14.

Replacing Re with Re + RS would produce the more general form.
156

15.

We have qualitatively compared Q factors of eight metallized and 14 doped Si
beams measured in different experimental runs, spanning the indicated frequency
range.

16.

X. Liu, E. J. Thompson, B. E. White, and R. O. Pohl Low-temperature internal
friction in metal films and in plastically deformed bulk aluminum. Phys. Rev. B
59, 11767 (1999).

17.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C.
Stipe, and D. Rugar Quality factors in micron- and submicron-thick cantilevers.
J. Microelectromech. Syst. 9, 117 (2000).

157

PHASE NOISE OF
NANOELECTROMECHANICAL SYSTEMS

Thesis by
Ya-Tang Yang

In Partial Fulfillment of the Requirements for the
Degree of

Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
2007
(Defended August 9, 2004)

Ya-Tang Yang

iii

Dedicated to
my family